research article classical ergodicity and modern portfolio ...research article classical ergodicity...

Research ArticleClassical Ergodicity and Modern Portfolio Theory

Geoffrey Poitras and John Heaney

Beedie School of Business Simon Fraser University Vancouver BC Canada V5A lS6

Correspondence should be addressed to Geoffrey Poitras poitrassfuca

Received 6 February 2015 Accepted 5 April 2015

Academic Editor Wing K Wong

Copyright copy 2015 G Poitras and J Heaney This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

What role have theoretical methods initially developed in mathematics and physics played in the progress of financial economicsWhat is the relationship between financial economics and econophysics What is the relevance of the ldquoclassical ergodicityhypothesisrdquo tomodernportfolio theoryThis paper addresses these questions by reviewing the etymology andhistory of the classicalergodicity hypothesis in 19th century statistical mechanics An explanation of classical ergodicity is provided that establishes aconnection to the fundamental empirical problem of using nonexperimental data to verify theoretical propositions in modernportfolio theoryThe role of the ergodicity assumption in the ex postex ante quandary confronting modern portfolio theory is alsoexamined

ldquoAs the physicist builds models of the movement of matter in a frictionless environment the economist builds models wherethere are no institutional frictions to the movement of stock pricesrdquo

E Elton and M Gruber Modern Portfolio Theory and Investment Analysis (1984 p 273)

1 Introduction

At least since Markowitz [1] initiated modern portfoliotheory (MPT) it has often been maintained that the tradeoffbetween systematic risk and expected return is the mostimportant theoretical element of financial economics forexample Campbell [2] Extending Mirowski [3] the staticequilibrium methods used to develop propositions in MPTsuch as the capital asset pricing model can be traced tomathematical concepts developed from the deterministicldquorational mechanicsrdquo approach to 19th century physics Inthe years since Markowitz [1] financial economics has alsoadopted alternative mathematical methods frommore recentcontributions to physics especially the diffusion processesemployed by Black and Scholes [4] to determine optionpricesThe emergence of econophysics during the last decadeof the twentieth century for example Roehner [5] andJovanovic and Schinckus [6] has provided a variety oftheoretical and empirical methods adapted from physicsranging from statistical mechanics to chaos theory to analyzefinancial phenomena Yet despite considerable overlap inmethod contributions to econophysics have gained limited

attention in financial economics In contrast econophysi-cists generally consider financial economics to be primarilyconcerned with a core theory that is inconsistent with theempirical orientation of physical theory

Physical theory has evolved considerably from the con-strained optimization static equilibrium approach of rationalmechanics which underpins MPT In detailing historicaldevelopments in physics since the 19th century it is conven-tional to jump from the determinism of rational mechanicsto quantum mechanics to recent developments in chaostheory overlooking the relevance of the initial steps towardmodeling stochastic behavior of physical phenomena byLudwig Boltzmann (1844ndash1906) James Maxwell (1831ndash1879)and Josiah Gibbs (1839ndash1903) As such there is a point ofdemarcation between the intellectual prehistories of MPTand econophysics that can arguably be traced to the debateover energistics around the end of the 19th centuryWhile theevolution of physics after energistics involved the introduc-tion and subsequent stochastic generalization of ergodic con-cepts fueled by the emergence of MPT following Markowitzfinancial economics incorporated ergodicity into empiricalmethods aimed at generalizing and testing the capital asset

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2015 Article ID 737905 17 pageshttpdxdoiorg1011552015737905

2 Chinese Journal of Mathematics

pricing model and other elements of MPT1 Significantlystochastic generalization of the static equilibrium approachof MPT required the adoption of a restricted class of ergodicprocesses that is ldquotime reversiblerdquo probabilistic modelsespecially the unimodal likelihood functions associated withcertain stationary distributions

In contrast from the early ergodic models of Boltzmannto the fractals and chaos theory of Mandlebrot physicshas employed a wider variety of ergodic and nonergodicstochastic models aimed at capturing key empirical charac-teristics of various physical problems at hand Such modelstypically have a mathematical structure that varies fromthe constrained optimization techniques underpinningMPTrestricting the straightforward application of many physicalmodels Yet the demarcation between the use of ergodicnotions in physics and financial economics was blurred sub-stantively by the introduction of diffusion process techniquesto solve contingent claims valuation problems Followingcontributions by Sprenkle [7] and Samuelson [8] Blackand Scholes [4] and Merton [9] provided an empiricallyviable method of using diffusion methods to determineusing Itorsquos lemma a partial differential equation that canbe solved for an option price Use of Itorsquos lemma to solvestochastic optimization problems is now commonplace infinancial economics for example Brennan and Schwartz [10]including the continuous time generalizations of MPT forexample Epstein and Ji [11] In spite of the considerableprogression of certain mathematical techniques employed inphysics into financial economics overcoming the difficultiesof applying the wide range of models developed for physicalsituations to fit the empirical properties of financial data isstill a central problem confronting econophysics

Schinckus [12 p 3816] accurately recognizes that thepositivist philosophical foundation of econophysics dependsfundamentally on empirical observation ldquoThe empiricistdimension is probably the first positivist feature of econo-physicsrdquo Following McCauley [13] and others this con-cern with empiricism often focuses on the identification ofmacrolevel statistical regularities that are characterized bythe scaling laws identified by Mandelbrot [14] and Man-dlebrot and Hudson [15] for financial data Unfortunatelythis empirically driven ideal is often confounded by theldquononrepeatablerdquo experiment that characterizesmost observedeconomic and financial data There is quandary posed byhaving only a single observed ex post time path to estimatethe distributional parameters for the ensemble of ex antetime paths needed to make decisions involving future valuesof financial variables In contrast to natural sciences suchas physics in the human sciences there is no assurancethat ex post statistical regularity translates into ex ante fore-casting accuracy Resolution of this quandary highlights theusefulness of employing a ldquophenomenologicalrdquo approach tomodeling stochastic properties of financial variables relevantto MPT

To this end this paper provides an etymology andhistory of the ldquoclassical ergodicity hypothesisrdquo in 19th cen-tury statistical mechanics Subsequent use of ergodicity infinancial economics in general and MPT in particularis then examined A modern interpretation of classical

ergodicity is provided that uses Sturm-Liouville theory amathematical method central to classical statistical mechan-ics to decompose the transition probability density of aone-dimensional diffusion process subject to regular upperand lower reflecting barriers This ldquoclassicalrdquo decompositiondivides the transition density of an ergodic process intoa possibly multimodal limiting stationary density whichis independent of time and initial condition and a powerseries of time and boundary dependent transient terms Incontrast empirical theory aimed at estimating relationshipsfrom MPT typically ignores the implications of the initialand boundary conditions that generate transient terms andfocuses on properties of a particular class of unimodal lim-iting stationary densities with finite parameters To illustratethe implications of the expanded class of ergodic processesavailable to econophysics properties of the bimodal quarticexponential stationary density are considered and used toassess the ability of the classical ergodicity hypothesis toexplain certain ldquostylized factsrdquo associated with the ex postexante quandary confronting MPT

2 A Brief History of Classical Ergodicity

The Encyclopedia of Mathematics [16] defines ergodic theoryas the ldquometric theory of dynamical systems the branch ofthe theory of dynamical systems that studies systems withan invariant measure and related problemsrdquo This moderndefinition implicitly identifies the birth of ergodic theorywith proofs of the mean ergodic theorem by von Neumann[17] and the pointwise ergodic theorem by Birkhoff [18]These early proofs have had significant impact in a widerange of modern subjects For example the notions ofinvariant measure and metric transitivity used in the proofsare fundamental to the measure theoretic foundation ofmodern probability theory [19 20] Building on a seminalcontribution to probability theory by Kolmogorov [21] inthe years immediately following it was recognized that theergodic theorems generalize the strong law of large numbersSimilarly the equality of ensemble and time averagesmdashtheessence of the mean ergodic theoremmdashis necessary to theconcept of a strictly stationary stochastic process Ergodictheory is the basis for themodern study of randomdynamicalsystems for example Arnold [22] In mathematics ergodictheory connects measure theory with the theory of transfor-mation groups This connection is important in motivatingthe generalization of harmonic analysis from the real line tolocally compact groups

From the perspective of modern mathematics statisticalphysics or systems theory Birkhoff [18] and von Neumann[17] are excellent starting points for a modern history ofergodic theory Building on the modern ergodic theoremssubsequent developments in these and related fields havebeen dramatic These contributions mark the solution to aproblem in statistical mechanics and thermodynamics thatwas recognized sixty years earlier when Ludwig Boltzmannintroduced the classical ergodic hypothesis to permit the the-oretical phase space average to be interchanged with themea-surable time average For the purpose of contrastingmethods

Chinese Journal of Mathematics 3

from physics and econophysics with those used in MPT theselection of the less formally correct and rigorous classicalergodic hypothesis of Boltzmann is a more auspicious begin-ning Problems of interest in mathematics are generated by arange of subjects such as physics chemistry engineering andbiology The formulation and solution of physical problemsin say statistical mechanics or particle physics will havemathematical features which are inapplicable or unnecessaryfor MPT For example in statistical mechanics points in thephase space are often multidimensional functions represent-ing the mechanical state of the system hence the desirabilityof a group-theoretic interpretation of the ergodic hypothesisFrom the perspective of MPT such complications are largelyirrelevant The history of classical ergodic theory capturesthe etymology and basic physical interpretation providing amore revealing prehistory of the relevant MPT mathematicsThis arguably more revealing prehistory begins with theformulation of theoretical problems that von Neumann andBirkhoff were later able to solve

Mirowski [3] [23 esp ch 5] establishes the importance of19th century physics in the development of the neoclassicaleconomic system advanced by W Stanley Jevons (1835ndash1882) and Leon Walras (1834ndash1910) during the marginalistrevolution of the 1870s Being derived using principles fromneoclassical economic theory MPT also inherited essentialfeatures of mid-19th century physics deterministic rationalmechanics conservation of energy and the nonatomisticcontinuum view of matter that inspired the energeticsmovement later in the 19th century2 More precisely fromneoclassical economicsMPT inherited a variety of static equi-librium techniques and tools such as mean-variance utilityfunctions and constrained optimization As such failingsof neoclassical economics identified by econophysicists alsoapply to central propositions of MPT Included in the failingsis an overemphasis on theoretical results at the expense ofidentifying models that have greater empirical validity forexample Roehner [5] and Schinckus [12]

It was during the transition from rational to statisticalmechanics during the last third of the 19th century thatBoltzmannmade contributions leading to the transformationof theoretical physics from the microscopic mechanisticmodels of Rudolf Clausius (1835ndash1882) and James Maxwellto the macroscopic probabilistic theories of Josiah Gibbs andAlbert Einstein (1879ndash1955)3 Coming largely after the startof the marginalist revolution in economics this fundamentaltransformation in theoretical physics had little impact onthe progression of financial economics until the appearanceof diffusion equations in contributions on continuous timefinance that started in the 1960s and culminated in Black andScholes [4] The deterministic mechanics of the energisticapproach was well suited to the axiomatic formalization ofneoclassical economic theory which culminated in the fol-lowing the von Neumann and Morgenstern expected utilityapproach to modeling uncertainty the Bourbaki inspiredArrow-Debreu general equilibrium theory for exampleWeintraub [24] and ultimately MPT In turn empiricalestimation and the subsequent extension of static equilibriumMPT results to continuous time were facilitated by theadoption of a narrow class of ergodic processes

Having descended from the deterministic rationalmechanics of mid-19th century physics defining works ofMPT do not capture the probabilistic approach to modelingsystems initially introduced by Boltzmann and furtherclarified by Gibbs4 Mathematical problems raised byBoltzmann were subsequently solved using tools introducedin a string of later contributions by the likes of the Ehrenfestsand Cantor in set theory Gibbs and Einstein in physicsLebesgue in measure theory Kolmogorov in probabilitytheory and Weiner and Levy in stochastic processesBoltzmann was primarily concerned with problems in thekinetic theory of gases formulating dynamic properties ofthe stationary Maxwell distribution the velocity distributionof gas molecules in thermal equilibrium Starting in 1871Boltzmann took this analysis one step further to determinethe evolution equation for the distribution function Themathematical implications of this classical analysis stillresonate in many subjects of the modern era The etymologyfor ldquoergodicrdquo begins with an 1884 paper by Boltzmannthough the initial insight to use probabilities to describe a gassystem can be found as early as 1857 in a paper by Clausiusand in the famous 1860 and 1867 papers by Maxwell5

The Maxwell distribution is defined over the velocity ofgas molecules and provides the probability for the relativenumber of molecules with velocities in a certain rangeUsing a mechanical model that involved molecular collisionMaxwell [25] was able to demonstrate that in thermalequilibrium this distribution of molecular velocities wasa ldquostationaryrdquo distribution that would not change shapedue to ongoing molecular collision Boltzmann aimed todetermine whether the Maxwell distribution would emergein the limit whatever the initial state of the gas In order tostudy the dynamics of the equilibrium distribution over timeBoltzmann introduced the probability distribution of therelative time a gas molecule has a velocity in a certain rangewhile still retaining the notion of probability for velocitiesof a relative number of gas molecules Under the classicalergodic hypothesis the average behavior of the macroscopicgas system which can objectively be measured over time canbe interchanged with the average value calculated from theensemble of unobservable and highly complex microscopicmolecular motions at a given point in time In the wordsof Weiner [26 p 1] ldquoBoth in the older Maxwell theory andin the later theory of Gibbs it is necessary to make somesort of logical transition between the average behavior of alldynamical systems of a given family or ensemble and thehistorical average of a single systemrdquo

3 Use of the Ergodic Hypothesis inFinancial Economics

At least since Samuelson [27] it has been recognized thatempirical theory and estimation in economics in generaland financial economics in particular relies heavily on theuse of specific unimodal stationary distributions associatedwith a particular class of ergodic processes As reflected inthe evolution of the concept in economics the specificationand implications of ergodicity have only developed gradually


The early presentation of ergodicity by Samuelson [27]involves the addition of a discrete Markov error term into thedeterministic cobweb model to demonstrate that estimatedforecasts of future values such as prices ldquoshould be lessvariable than the actual datardquo Considerable opaqueness aboutthe definition of ergodicity is reflected in the statement thata ldquolsquostablersquo stochastic process eventually forgets its past andtherefore in the far future can be expected to approach anergodic probability distributionrdquo [27 p 2] The connectionbetween ergodic processes and nonlinear dynamics that char-acterizes present efforts in economics goes unrecognizedfor example [27 p 1 5] While some explicit applicationsof ergodic processes to theoretical modeling in economicshave emerged since Samuelson [27] for example Horst andWenzelburger [28] and Dixit and Pindyck [29] financialeconometrics has produced the bulk of the contributions

Initial empirical estimation for the deterministic modelsof neoclassical economics proceeded with the addition of astationary usually Gaussian error term to produce a discretetime general linear model (GLM) leading to estimation usingordinary least squares or maximum likelihood techniques Inthe history of MPT such early estimations were associatedwith tests of the capital asset pricing model such as the ldquomar-ket modelrdquo for example Elton and Gruber [30] Iterationsand extensions of the GLM to deal with complications arisingin empirical estimates dominated early work in economet-rics for example Dhrymes [31] and Theil [32] leading toapplication of generalized least squares estimation techniquesthat encompassed autocorrelated and heteroskedastic errorterms Employing 119871

2vector space methods with stationary

Gaussian-based error term distributions ensured these earlystochastic models implicitly assumed ergodicityThe general-ization of this discrete time estimation approach to the class ofARCH andGARCH error termmodels by Engle andGrangerwas of such significance that a Nobel prize in economicswas awarded for this contribution for example Engle andGranger [33] By modeling the evolution of the volatility thisapproach permitted a limited degree of nonlinearity to bemodeled providing a substantively better fit of MPT modelsto observed financial time series for example Beaulieu et al[34]

The emergence of ARCH GARCH and related empiricalmodels was part of a general trend toward the use of inductivemethods in economics often employing discrete linear timeseries methods tomodel transformed economic variables forexample Hendry [35] At least since Dickey and Fuller [36]it has been recognized that estimates of univariate time seriesmodels for many financial times series reveals evidence ofldquonon-stationarityrdquo A number of approaches have emergedto deal with this apparent empirical quandary6 In particu-lar transformation techniques for time series models havereceived considerable attention Extension of the Box-Jenkinsmethodology led to the concept of economic time seriesbeing I(0)mdashstationary in the levelmdashand I(1)mdashnonstationaryin the level but stationary after first differencing Two I(1)economic variables could be cointegrated if differencing thetwo series produced an I(0) process for example Hendry[35] Extending early work on distributed lags long memory

processes have also been employed where the time series isonly subject to fractional differencing Significantly recentcontributions on Markov switching processes and exponen-tial smooth transition autoregressive processes have demon-strated the ldquopossibility that nonlinear ergodic processes canbe misinterpreted as unit root nonstationary processesrdquo [37p 620] Bonomo et al [38] illustrates the recent applicationof Markov switching processes in estimating the asset pricingmodels of MPT

The conventional view of ergodicity in economics ingeneral and financial economics in particular is reflected byHendry [35 p 100] ldquoWhether economic reality is an ergodicprocess after suitable transformation is a deep issuerdquo whichis difficult to analyze rigorously As a consequence in thelimited number of instances where ergodicity is examinedin economics a variety of different interpretations appearIn contrast the ergodic hypothesis in classical statisticalmechanics is associated with the more physically transparentkinetic gas model than the often technical and targetedconcepts of ergodicity encountered in modern economicsin general and MPT in particular For Boltzmann theclassical ergodic hypothesis permitted the unobserved com-plex microscopic interactions of individual gas moleculesto obey the second law of thermodynamics a concept thathas limited application in economics7 Despite differences inphysical interpretation there is similarity to the problem ofmodeling ldquomacroscopicrdquo financial variables such as commonstock prices foreign exchange rates ldquoassetrdquo prices or interestrates By construction when it is not possible to derive atheory for describing and predicting empirical observationsfrom known first principles about the (microscopic) rationalbehavior of individuals and firms By construction thisinvolves a phenomenological approach to modeling8

Even though the formal solutions proposed were inad-equate by standards of modern mathematics the thermo-dynamic model introduced by Boltzmann to explain thedynamic properties of the Maxwell distribution is a peda-gogically useful starting point to develop the implicationsof ergodicity for MPT To be sure von Neumann [17] andBirkhoff [18] correctly specify ergodicity using Lebesgue inte-gration an essential analytical tool unavailable to Boltzmannbut the analysis is too complex to be of much value to allbut the most mathematically specialized economists Thephysical intuition of the kinetic gas model is lost in thegenerality of the results Using Boltzmann as a starting pointthe large number of mechanical and complex molecularcollisions could correspond to the large number of micro-scopic atomistic liquidity providers and traders interactingto determine the macroscopic financial market price In thiscontext it is variables such as the asset price or the interestrate or the exchange rate or some combination that isbeingmeasured over time and ergodicity would be associatedwith the properties of the transition density generating themacroscopic variables Ergodicity can fail for a number ofreasons and there is value in determining the source of thefailure In this vein there are two fundamental difficultiesassociated with the classical ergodicity hypothesis in Boltz-mannrsquos statistical mechanics reversibility and recurrencethat are largely unrecognized in financial economics9


Halmos [39 p 1017] is a helpful starting point to sortout the differing notions of ergodicity that are of relevanceto the issues at hand ldquoThe ergodic theorem is a statementabout a space a function and a transformationrdquo In mathe-matical terms ergodicity or ldquometric transitivityrdquo is a propertyof ldquoindecomposablerdquo measure preserving transformationsBecause the transformation acts on points in the space thereis a fundamental connection to the method of measuringrelationships such as distance or volume in the space In vonNeumann [17] and Birkhoff [18] this is accomplished usingthe notion of Lebesgue measure the admissible functionsare either integrable (Birkhoff) or square integrable (vonNeumann) In contrast to say statistical mechanics wherespaces and functions account for the complex physicalinteraction of large numbers of particles economic theoriessuch as MPT usually specify the space in a mathematicallyconvenient fashion For example in the case where there is asingle random variable then the space is ldquosuperfluousrdquo [20p 182] as the random variable is completely described bythe distribution Multiple random variables can be handledby assuming the random variables are discrete with finitestate spaces In effect conditions for an ldquoinvariant measurerdquoare assumed in MPT in order to focus attention on ldquofindingand studying the invariant measuresrdquo [22 p 22] wherein the terminology of financial econometrics the invariantmeasure usually corresponds to the stationary distribution orlikelihood function

The mean ergodic theorem of von Neumann [17] pro-vides an essential connection to the ergodicity hypothesisin financial econometrics It is well known that in theHilbert and Banach spaces common to econometric workthe mean ergodic theorem corresponds to the strong lawof large numbers In statistical applications where strictlystationary distributions are assumed the relevant ergodictransformation 119871lowast is the unit shift operator 119871lowastΨ[119909(119905)] =

Ψ[119871

lowast119909(119905)] = Ψ[119909(119905 + 1)] [(119871lowast)119896]Ψ[119909(119905)] = Ψ[119909(119905 +

119896)] and (119871

lowast)

minus119896Ψ[119909(119905)] = Ψ[119909(119905 minus 119896)] with 119896 being an

integer and Ψ[119909] the strictly stationary distribution for 119909that in the strictly stationary case is replicated at each 11990510Significantly this reversible transformation is independentof initial time and state Because this transformation canbe achieved by imposing strict stationarity on Ψ[119909] 119871lowast willonly work for certain ergodic processes In effect the ergodicrequirement that the transformation is measure preserving isweaker than the strict stationarity of the stochastic processsufficient to achieve 119871lowast The practical implications of thereversible ergodic transformation 119871lowast are described by David-son [40 p 331] ldquoIn an economic world governed entirely by[time reversible] ergodic processes economic relationshipsamong variables are timeless or ahistoric in the sense that thefuture is merely a statistical reflection of the past [sic]rdquo11

Employing conventional econometrics in empirical stud-ies MPT requires that the real world distribution for 119909(119905) forexample the asset return is sufficiently similar to those forboth 119909(119905 + 119896) and 119909(119905 minus 119896) that is the ergodic transformation119871

lowast is reversible The reversibility assumption is systemic inMPT appearing in the use of long estimation periods to deter-mine important variables such as the ldquoequity risk premiumrdquo

There is a persistent belief that increasing the length orsampling frequency of a financial time series will improvethe precision of a statistical estimate for example Dimsonet al [41] Similarly focus on the tradeoff between ldquorisk andreturnrdquo requires the use of unimodal stationary densities fortransformed financial variables such as the rate of returnThe impact of initial and boundary conditions on financialdecision making is generally ignored The inconsistency ofreversible processes with key empirical facts such as theasymmetric tendency for downdrafts in prices to be moresevere than upswings is ignored in favor of adhering toldquoreversiblerdquo theoretical models that can be derived from firstprinciples associated with constrained optimization tech-niques for example Constantinides [42]

4 A Phenomenological Interpretation ofClassical Ergodicity

In physics phenomenology lies at the intersection of theoryand experiment Theoretical relationships between empiri-cal observations are modeled without deriving the theorydirectly from first principles for example Newtonrsquos laws ofmotion Predictions based on these theoretical relationshipsare obtained and compared to further experimental datadesigned to test the predictions In this fashion new theoriesthat can be derived from first principles are motivated Con-fronted with nonexperimental data for important financialvariables such as common stock prices interest rates andthe like financial economics has developed some theoreticalmodels that aim to fit the ldquostylized factsrdquo of those variablesIn contrast the MPT is initially derived directly from theldquofirst principlesrdquo of constrained expected utility maximizingbehavior by individuals and firms Given the difficulties ineconomics of testing model predictions with ldquonewrdquo experi-mental data physics and econophysics have the potential toprovide a rich variety of mathematical techniques that canbe adapted to determiningmathematical relationships amongfinancial variables that explain the ldquostylized factsrdquo of observednonexperimental data12

The evolution of financial economics from the deter-ministic models of neoclassical economics to more modernstochastic models has been incremental and disjointed Thepreference for linear models of static equilibrium relation-ships has restricted the application of theoretical frameworksthat capture more complex nonlinear dynamics for exam-ple chaos theory truncated Levy processes Yet importantfinancial variables have relatively innocuous sample pathscompared to some types of variables encountered in physicsThere is an impressive range of mathematical and statisticalmodels that seemingly could be applied to almost any physi-cal or financial situation If the process can be verbalized thena model can be specified This begs the following questionsare there transformations ergodic or otherwise that capturethe basic ldquostylized factsrdquo of observed financial data Is therandom instability in the observed sample paths identifiedin say stock price time series consistent with the ex antestochastic bifurcation of an ergodic process for exampleChiarella et al [43] In the bifurcation case the associated


ex ante stationary densities are bimodal and irreversible asituation where the mean calculated from past values of asingle nonexperimental ex post realization of the process isnot necessarily informative about the mean for future values

Boltzmann was concerned with demonstrating that theMaxwell distribution emerged in the limit as 119905 rarr infin forsystems with large numbers of particles The limiting processfor 119905 requires that the system run long enough that theinitial conditions do not impact the stationary distributionAt the time two fundamental criticisms were aimed at thisgeneral approach reversibility and recurrence In the contextof financial time series reversibility relates to the use of pastvalues of the process to forecast future values Recurrencerelates to the properties of the long run average whichinvolves the ability and length of time for an ergodic processto return to its stationary state For Boltzmann both thesecriticisms have roots in the difficulty of reconciling the secondlaw of thermodynamics with the ergodicity hypothesis UsingSturm-Liouville methods it can be shown that classicalergodicity requires the transition density of the process tobe decomposable into the sum of a stationary density anda mean zero transient term that captures the impact of theinitial condition of the system on the individual sample pathsirreversibility relates to properties of the stationary densityand nonrecurrence to the behavior of the transient term

Because the particlemovements in a kinetic gasmodel arecontained within an enclosed system for example a verticalglass tube classical Sturm-Liouville (S-L) methods can beapplied to obtain solutions for the transition densities Theseclassical results for the distributional implications of impos-ing regular reflecting boundaries on diffusion processes arerepresentative of the modern phenomenological approach torandom systems theory which ldquostudies qualitative changes ofthe densites [sic] of invariant measures of the Markov semi-group generated by random dynamical systems induced bystochastic differential equationsrdquo [44 p 27]13 Because the ini-tial condition of the system is explicitly recognized ergodicityin these models takes a different form than that associatedwith the unit shift transformation of unimodal stationarydensities typically adopted in financial economics in generaland MPT in particular The ergodic transition densities arederived as solutions to the forward differential equationassociated with one-dimensional diffusions The transitiondensities contain a transient term that is dependent on theinitial condition of the system and boundaries imposed onthe state space Path dependence that is irreversibility canbe introduced by employingmultimodal stationary densities

The distributional implications of boundary restrictionsderived by modeling the random variable as a diffusionprocess subject to reflecting barriers have been studied formany years for example Feller [45] The diffusion processframework is useful because it imposes a functional structurethat is sufficient for known partial differential equation(PDE) solution procedures to be used to derive the relevanttransition probability densitiesWong [46] demonstrated thatwith appropriate specification of parameters in the PDE thetransition densities for popular stationary distributions suchas the exponential uniform and normal distributions canbe derived using classical S-L methods The S-L framework

provides sufficient generality to resolve certain empiricaldifficulties arising from key stylized facts observed in thenonexperimental time series from financial economics Moregenerally the framework suggests a method of allowingMPT to encompass the nonlinear dynamics of diffusion pro-cesses In other words within the more formal mathematicalframework of classical statistical mechanics it is possibleto reformulate the classical ergodicity hypothesis to permita useful stochastic generalization of theories in financialeconomics such as MPT

The use of the diffusion model to represent the nonlineardynamics of stochastic processes is found in a wide rangeof subjects Physical restrictions such as the rate of observedgenetic mutation in biology or character of heat diffusion inengineering or physics often determine the specific formal-ization of the diffusion model Because physical interactionscan be complexmathematical results for diffusionmodels arepitched at a level of generality sufficient to cover such cases14Such generality is usually not required in financial economicsIn this vein it is possible to exploit mathematical propertiesof bounded state spaces and one-dimensional diffusionsto overcome certain analytical problems that can confrontcontinuous time Markov solutions The key construct inthe S-L method is the ergodic transition probability densityfunction 119880 which is associated with the random (financial)variable 119909 at time 119905 (119880 = 119880[119909 119905 | 119909

0]) that follows a regular

time homogeneous diffusion process While it is possible toallow the state space to be an infinite open interval 119868

119900= (119886 119887

infin le 119886 lt 119887 le infin) a finite closed interval 119868119888= [119886 119887

minusinfin lt 119886 lt 119887 lt +infin] or the specific interval 119868119904= [0 =

119886 lt 119887 lt infin) are applicable to financial variables15 Assumingthat 119880 is twice continuously differentiable in 119909 and once in 119905and vanishes outside the relevant interval then 119880 obeys theforward equation (eg [47 p 102ndash4])

120597

2

120597119909

2 119861 [119909]119880 minus120597

120597119909

119860 [119909]119880 =

120597119880

120597119905

(1)

where119861[119909] (=(12)1205902[119909] gt 0) is the one half the infinitesimalvariance and 119860[119909] the infinitesimal drift of the process 119861[119909]is assumed to be twice and 119860[119909] once continuously differ-entiable in 119909 Being time homogeneous this formulationpermits state but not time variation in the drift and varianceparameters

If the diffusion process is subject to upper and lowerreflecting boundaries that are regular and fixed (minusinfin lt 119886 lt

119887 lt infin) the classical ldquoSturm-Liouville problemrdquo involvessolving (1) subject to the separated boundary conditions16

120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886]119880 [119886 119905] = 0 (2)

120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887]119880 [119887 119905] = 0 (3)

And the initial condition is as follows

119880 [119909 0] = 119891 [1199090] where int119887

119886

119891 [1199090] = 1 (4)


and 119891[1199090] is the continuous density function associated with1199090 where 119886 le 1199090 le 119887 When the initial starting value 119909

0

is known with certainty the initial condition becomes theDirac delta function 119880[119909 0] = 120575[119909 minus 119909

0] and the resulting

solution for119880 is referred to as the ldquoprincipal solutionrdquoWithinthe framework of the S-L method a stochastic process hasthe property of classical ergodicity when the transition densitysatisfies the following17

lim119905rarrinfin

119880 [119909 119905 | 1199090] = int119887

119886

119891 [1199090] 119880 [119909 119905 | 1199090] 1198891199090

= Ψ [119909]

(5)

Important special cases occur for the principal solution(119891[1199090] = 120575[119909 minus 119909

0]) and when 119891[119909

0] is from a specific

class such as the Pearson distributions To be ergodic thetime invariant stationary density Ψ[119909] is not permitted toldquodecomposerdquo the sample space with a finite number ofindecomposable subdensities each of which is time invariantSuch irreversible processes are not ergodic even though eachof the subdensities could be restricted to obey the ergodictheorem To achieve ergodicity a multimodal stationarydensity can be used instead of decomposing the sample spaceusing subdensities with different means In turn multimodalirreversible ergodic processes have the property that themeancalculated from past values of the process are not necessarilyinformative enough about the modes of the ex ante densitiesto provide accurate predictions

In order to more accurately capture the ex ante propertiesof financial time series there are some potentially restrictivefeatures in the classical S-L framework that can be identifiedFor example time homogeneity of the process eliminatesthe need to explicitly consider the location of 119905

018 Time

homogeneity is a property of119880 and as such is consistent withldquoahistoricalrdquo MPT In the subclass of 119880 denoted as 119880lowasta time homogeneous and reversible stationary distributiongoverns the dynamics of 119909(119905) Significantly while 119880 is timehomogeneous there are some 119880 consistent with irreversibleprocesses A relevant issue for econophysicists is to deter-mine which conceptmdashtime homogeneity or reversibilitymdashis inconsistent with economic processes that capture col-lapsing pricing conventions in asset markets liquidity trapsin money markets and collapse induced structural shiftsin stock markets In the S-L framework the initial state ofthe system (119909

0) is known and the ergodic transition density

provides information about how a given point 1199090shifts 119905

units along a trajectory19 In contrast applications in financialeconometrics employ the strictly stationary 119880lowast where thelocation of 119909

0is irrelevant while 119880 incorporates 119909

0as an

initial condition associated with the solution of a partialdifferential equation

5 Density Decomposition Results20

In general solving the forward equation (1) for 119880 subjectto (2) (3) and some admissible form of (4) is difficultfor example Feller [45] and Risken [48] In such circum-stances it is expedient to restrict the problem specification

to permit closed form solutions for the transition densityto be obtained Wong [46] provides an illustration of thisapproach The PDE (1) is reduced to an ODE by onlyconsidering the strictly stationary distributions arising fromthe Pearson system Restrictions on the associated Ψ[119909] areconstructed by imposing the fundamental ODE condition forthe unimodal Pearson system of distributions

119889Ψ [119909]

119889119909

=

1198901119909 + 11989001198892119909

2+ 1198891119909 + 1198890

Ψ [119909] (6)

The transition probability density 119880 for the ergodic processcan then be reconstructed by working back from a specificclosed form for the stationary distribution using knownresults for the solution of specific forms of the forwardequation In this procedure the 119889

0 1198891 1198892 1198900 and 119890

1in the

Pearson ODE are used to specify the relevant parametersin (1) The 119880 for important stationary distributions that fallwithin the Pearson system such as the normal beta central119905 and exponential can be derived by this method

The solution procedure employed byWong [46] dependscrucially on restricting the PDE problem sufficiently to applyclassical S-L techniques Using S-L methods various studieshave generalized the set of solutions for 119880 to cases wherethe stationary distribution is not a member of the Pearsonsystem or 119880 is otherwise unknown for example Linetsky[49] In order to employ the separation of variables techniqueused in solving S-L problems (1) has to be transformed intothe canonical form of the forward equation To do this thefollowing function associated with the invariant measure isintroduced

119903 [119909] = 119861 [119909] exp [minusint119909

119886

119860 [119904]

119861 [119904]

119889119904] (7)

Using this function the forward equation can be rewritten inthe form

1119903 [119909]

120597

120597119909

119901 [119909]

120597119880

120597119909

+ 119902 [119909]119880 =

120597119880

120597119905

(8)

where119901 [119909] = 119861 [119909] 119903 [119909]

119902 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(9)

Equation (8) is the canonical form of (1) The S-L problemnow involves solving (8) subject to appropriate initial andboundary conditions

Because the methods for solving the S-L problem areODE-based some method of eliminating the time deriva-tive in (1) is required Exploiting the assumption of timehomogeneity the eigenfunction expansion approach appliesseparation of variables permitting (8) to be specified as

119880 [119909 119905] = 119890

minus120582119905120593 [119909]

(10)

where120593[119909] is only required to satisfy the easier-to-solveODE

1119903 [119909]

119889

119889119909

[119901 [119909]

119889120593

119889119909

]+ [119902 [119909] + 120582] 120593 [119909] = 0 (11015840)


Transforming the boundary conditions involves substitutionof (10) into (2) and (3) and solving to get

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)

Significant analytical advantages are obtained by making theS-L problem ldquoregularrdquo which involves assuming that [119886 119887] is aclosed interval with 119903[119909]119901[119909] and 119902[119909] being real valued and119901[119909] having a continuous derivative on [119886 119887] and 119903[119909] gt 0119901[119909] gt 0 at every point in [119886 119887] ldquoSingularrdquo S-L problems arisewhere these conditions are violated due to say an infinitestate space or a vanishing coefficient in the interval [119886 119887]The separated boundary conditions (2) and (3) ensure theproblem is self-adjoint [50 p 91]

The classical S-L problem of solving (8) subject to theinitial and boundary conditions admits a solution only forcertain critical values of 120582 the eigenvalues Further since(1) is linear in 119880 the general solution for (10) is given by alinear combination of solutions in the form of eigenfunctionexpansions Details of these results can be found in Hille[51 ch 8] Birkhoff and Rota [52 ch 10] and Karlin andTaylor [53] When the S-L problem is self-adjoint and regularthe solutions for the transition probability density can besummarized in the following (see the appendix for proof)

Proposition 1 (ergodic transition density decomposition)The regular self-adjoint Sturm-Liouville problem has an infi-nite sequence of real eigenvalues 0 = 120582

0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)

To each eigenvalue there corresponds a unique eigenfunction120593

119899equiv 120593

119899[119909] Normalization of the eigenfunctions produces

120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)

The 120595119899[119909] eigenfunctions form a complete orthonormal system

in 1198712[119886 119887] The unique solution in 119871

2[119886 119887] to (1) subject to

the boundary conditions (2)-(3) and initial condition (4) is ingeneral form

119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)

Given this the transition probability density function for 119909at time 119905 can be reexpressed as the sum of a stationarylimiting equilibrium distribution associated with the 120582

0= 0

eigenvalue that is linearly independent of the boundaries and

a power series of transient terms associated with the remainingeigenvalues that are boundary and initial condition dependent

119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)

Using the specifications of 120582119899 119888119899 and 120595

119899 the properties of

119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)

This proposition provides the general solution to theregular self-adjoint S-L problem of deriving 119880 when theprocess is subject to regular reflecting barriers Taking thelimit as 119905 rarr infin in (15) it follows from (16) and (17)that the transition density of the stochastic process satisfiesthe classical ergodic property Considerable effort has beengiven to determining the convergence behavior of differentprocesses The distributional impact of the initial conditionsand boundary restrictions enter through 119879[119909 119905 | 1199090] Fromthe restrictions on 119879[119909 119905 | 1199090] in (17) the total mass ofthe transient term is zero so the mean ergodic theorem stillapplies The transient only acts to redistribute the mass ofthe stationary distribution thereby causing a change in shapewhich can impact the ex ante calculation of the expectedvalue The specific degree and type of alteration depends onthe relevant assumptions made about the parameters and ini-tial functional forms Significantly stochastic generalizationof static and deterministic MPT almost always ignores theimpact of transients by only employing the limiting stationarydistribution component

The theoretical advantage obtained by imposing regularreflecting barriers on the diffusion state space for the forwardequation is that an ergodic decomposition of the transitiondensity is assured The relevance of bounding the state spaceand imposing regular reflecting boundaries can be illustratedby considering the well known solution (eg [54 p 209]) for119880 involving a constant coefficient standard normal variate119884(119905) = (119909 minus 119909

0minus 120583119905120590) over the unbounded state space

119868

119900= (infin le 119909 le infin) In this case the forward equation

(1) reduces to (12)12059721198801205971198842 = 120597119880120597119905 By evaluating thesederivatives it can be verified that the principal solution for119880is

119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)

exp[minus(119909 minus 1199090 minus 120583119905)

2

21205902119905] (19)


and as 119905 rarr minusinfin or 119905 rarr +infin then 119880 rarr 0 andthe stochastic process is nonergodic because it does notpossess a nontrivial stationary distributionThemean ergodictheorem fails if the process runs long enough then 119880 willevolve to where there is no discernible probability associatedwith starting from 119909

0and reaching the neighborhood of a

given point 119909 The absence of a stationary distribution raisesa number of questions for example whether the processhas unit roots Imposing regular reflecting boundaries is acertain method of obtaining a stationary distribution and adiscrete spectrum [55 p 13] Alternative methods such asspecifying the process to admit natural boundaries where theparameters of the diffusion are zero within the state spacecan give rise to continuous spectrum and raise significantanalytical complexities At least since Feller [45] the searchfor useful solutions including those for singular diffusionproblems has produced a number of specific cases of interestHowever without the analytical certainty of the classical S-Lframework analysis proceeds on a case by case basis

One possible method of obtaining a stationary distribu-tion without imposing both upper and lower boundaries is toimpose only a lower (upper) reflecting barrier and constructthe stochastic process such that positive (negative) infinityis nonattracting for example Linetsky [49] and Aıt-Sahalia[56] This can be achieved by using a mean-reverting driftterm In contrast Cox and Miller [54 p 223ndash5] use theBrownianmotion constant coefficient forward equationwith119909

0gt 0119860[119909] = 120583 lt 0 and119861[119909] = (12)1205902 subject to the lower

reflecting barrier at 119909 = 0 given in (2) to solve for both the119880 and the stationary density The principal solution is solvedusing the ldquomethod of imagesrdquo to obtain

119880 [119909 119905 | 1199090] =1

120590radic2120587119905

expminus((119909 minus 1199090 minus 120583119905)

2

21205902119905)

+ expminus(41199090120583119905 minus (119909 minus 1199090 minus 120583119905)

2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)

where 119873[119909] is the cumulative standard normal distributionfunction Observing that 119860[119909] = 120583 gt 0 again produces 119880 rarr

0 as 119905 rarr +infin the stationary density for 119860[119909] = 120583 lt 0 hasthe Maxwell form

Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)

Though 1199090does not enter the solution combined with the

location of the boundary at 119909 = 0 it does implicitly imposethe restriction 119909 gt 0 From the Proposition 119879[119909 119905 | 1199090] canbe determined as 119880[119909 119905 | 119909

0] minus Ψ[119909]

Following Linetsky [49] Veerstraeten [57] and othersthe analytical procedure used to determine 119880 involvesspecifying the parameters of the forward equation and the

boundary conditions and then solving for Ψ[119909] and 119879[119909 119905 |1199090] Wong [46] uses a different approach initially selectinga stationary distribution and then solving for 119880 using therestrictions of the Pearson system to specify the forwardequation In this approach the functional form of the desiredstationary distribution determines the appropriate boundaryconditions While application of this approach has beenlimited to the restricted class of distributions associated withthe Pearson system it is expedient when a known stationarydistribution such as the standard normal distribution is ofinterest More precisely let

Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868

119900= (minusinfinlt119909ltinfin)

(22)

In this case the boundaries of the state space are nonat-tracting and not regular Solving the Pearson equation gives119889Ψ[119909]119889119909 = minus119909Ψ[119909] and a forward equation of theOU form

120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)

FollowingWong [46 p 268] Mehlerrsquos formula can be used toexpress the solution for 119880 as

119880 [119909 119905 | 1199090]

=

1

radic2120587 (1 minus 119890minus2119905)exp[

minus (119909 minus 1199090119890minus119905)

2

2 (1 minus 119890minus2119905)]

(24)

Given this as 119905 rarr minusinfin then 119880 rarr 0 and as 119905 rarr +infin then119880 achieves the stationary standard normal distribution

6 The Quartic Exponential Distribution

The roots of bifurcation theory can be found in the earlysolutions to certain deterministic ordinary differential equa-tions Consider the deterministic dynamics described by thepitchfork bifurcation ODE

119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588

1are the ldquonormalrdquo and ldquosplittingrdquo control

variables respectively (eg [58 59]) While 1205880has significant

information in a stochastic context this is not usually the casein the deterministic problem so 120588

0= 0 is assumed Given

this for 1205881le 0 there is one real equilibrium (119889119909119889119905 = 0)

solution to this ODE at 119909 = 0 where ldquoall initial conditionsconverge to the same final point exponentially fast with timerdquo[60 p 260] For 120588

1gt 0 the solution bifurcates into three

equilibrium solutions 119909 = 0 plusmnradic120588

1 one unstable and two

stable In this case the state space is split into two physicallydistinct regions (at 119909 = 0) with the degree of splittingcontrolled by the size of 120588

1 Even for initial conditions that

are ldquocloserdquo the equilibrium achieved will depend on thesign of the initial condition Stochastic bifurcation theory


extends this model to incorporate Markovian randomnessIn this theory ldquoinvariant measures are the random analoguesof deterministic fixed pointsrdquo [22 p 469] Significantlyergodicity now requires that the component densities thatbifurcate out of the stationary density at the bifurcation pointbe invariant measures for example Crauel et al [44 sec 3]As such the ergodic bifurcating process is irreversible in thesense that past sample paths (prior to the bifurcation) cannotreliably be used to generate statistics for the future values ofthe variable (after the bifurcation)

It is well known that the introduction of randomness tothe pitchfork ODE changes the properties of the equilibriumsolution for example [22 sec 92] It is no longer necessarythat the state space for the principal solution be determinedby the location of the initial condition relative to the bifur-cation point The possibility for randomness to cause somepaths to cross over the bifurcation point depends on the sizeof volatility of the process 120590 which measures the nonlinearsignal to white noise ratio Of the different approachesto introducing randomness (eg multiplicative noise) thesimplest approach to converting from a deterministic to astochastic context is to add a Weiner process (119889119882(119905)) to theODE Augmenting the diffusion equation to allow for 120590 tocontrol the relative impact of nonlinear drift versus randomnoise produces the ldquopitchfork bifurcationwith additive noiserdquo[22 p 475] which in symmetric form is

119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)

Applications in financial economics for example Aıt-Sahalia[56] refer to this diffusion process as the double well processWhile consistent with the common use of diffusion equationsin financial economics the dynamics of the pitchfork processcaptured by 119879[119909 119905 | 1199090] have been ldquoforgottenrdquo [22 p 473]

Models in MPT are married to the transition probabilitydensities associated with unimodal stationary distributionsespecially the class of Gaussian-related distributions Yet it iswell known that more flexibility in the shape of the stationarydistribution can be achieved using a higher order exponentialdensity for example Fisher [61] Cobb et al [62] and Craueland Flandoli [60] Increasing the degree of the polynomialin the exponential comes at the expense of introducing addi-tional parameters resulting in a substantial increase in theanalytical complexity typically defying a closed form solutionfor the transition densities However following Elliott [63] ithas been recognized that the solution of the associated regularS-L problem will still have a discrete spectrum even if thespecific form of the eigenfunctions and eigenvalues in119879[119909 119905 |1199090] is not precisely determined [64 sec 67] Inferences abouttransient stochastic behavior can be obtained by examiningthe solution of the deterministic nonlinear dynamics In thisprocess attention initially focuses on the properties of thehigher order exponential distributions

To this end assume that the stationary distribution is afourth degree or ldquogeneral quarticrdquo exponential

Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)

where 119870 is a constant determined such that the densityintegrates to one and 120573

4gt 021 Following Fisher [61]

the class of distributions associated with the general quarticexponential admits both unimodal and bimodal densities andnests the standard normal as a limiting case where 120573

4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(

radic2120587) The stationary

distribution of the bifurcating double well process is a specialcase of the symmetric quartic exponential distribution

Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)

where 120583 is the populationmean and the symmetry restrictionrequires 120573

1= 120573

3= 0 Such multimodal stationary densities

have received scant attention in financial economics ingeneral and in MPT in particular To see why the conditionon1205731is needed consider change of origin119883 = 119884minus120573

34120573

4 to

remove the cubic term from the general quartic exponential[65 p 480]

Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)

The substitution of 119910 for 119909 indicates the change of originwhich produces the following relations between coefficientsfor the general and specific cases

120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)

The symmetry restriction 120581 = 0 can only be satisfied if both120573

3and 120573

1= 0 Given the symmetry restriction the double

well process further requires minus120572 = 120574 = 120590 = 1 Solving forthe modes of Ψ[119910] gives plusmnradic|120572|(2120574) which reduces to plusmn1for the double well process as in Aıt-Sahalia [56 Figure 6Bp 1385]

As illustrated in Figure 1 the selection of 119886119894in the station-

ary density Ψ119894[119909] = 119870

119876expminus(0251199094 minus 051199092 minus 119886

119894119909) defines

a family of general quartic exponential densities where 119886119894

is the selected value of 120581 for that specific density22 Thecoefficient restrictions on the parameters 120572 and 120574 dictate thatthese values cannot be determined arbitrarily For examplegiven that 120573

4is set at 025 then for 119886

119894= 0 it follows that

120572 = 120573

2= 05 ldquoSlicing acrossrdquo the surface in Figure 1 at

119886

119894= 0 reveals a stationary distribution that is equal to the

double well density Continuing to slice across as 119886119894increases

in size the bimodal density becomes progressively moreasymmetrically concentrated in positive 119909 valuesThough thelocation of themodes does not change the amount of densitybetween the modes and around the negative mode decreasesSimilarly as 119886

119894decreases in size the bimodal density becomes

more asymmetrically concentrated in positive 119909 values


0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]

Figure 1 Family of stationary densities for Ψ

119894[119909] =

119870


119894119909) Each of the continuous values for

119886 signifies a different stationary density For example at 119886 = 0 thedensity is the double well density which symmetric about zero andwith modes at plusmn1

While the stationary density is bimodal over 119886119894120576minus1 1 for

|119886

119894| large enough the density becomes so asymmetric that

only a unimodal density appears For the general quarticasymmetry arises as the amount of the density surroundingeach mode (the subdensity) changes with 119886

119894 In this the

individual stationary subdensities have a symmetric shapeTo introduce asymmetry in the subdensities the reflectingboundaries at 119886 and 119887 that bound the state space for theregular S-L problem can be used to introduce positiveasymmetry in the lower subdensity and negative asymmetryin the upper subdensity

Following Chiarella et al [43] the stochastic bifurcationprocess has a number of features which are consistent withthe ex ante behavior of a securities market driven by acombination of chartists and fundamentalists Placed inthe context of the classical S-L framework because thestationary distributions are bimodal and depend on forwardparameters such as 120581 120572 120574 and 119886

119894in Figure 1 that are

not known on the decision date the risk-return tradeoffmodels employed in MPT are uninformative What use isthe forecast provided by 119864[119909(119879)] when it is known thatthere are other modes for 119909(119879) values that are more likelyto occur A mean estimate that is close to a bifurcationpoint would even be unstable The associated difficulty ofcalculating an expected return forecast or other statisticalestimate such as volatility from past data can be compoundedby the presence of transients that originate from boundariesand initial conditions For example the presence of a recentstructural break for example the collapse in global stockprices from late 2008 to early 2009 can be accounted forby appropriate selection of 119909

0 the fundamental dependence

of investment decisions on the relationship between 119909

0

and future (not past) performance is not captured by thereversible ergodic processes employed in MPT that ignoretransients arising from initial conditions Mathematical toolssuch as classical S-L methods are able to demonstrate thisfundamental dependence by exploiting properties of ex antebifurcating ergodic processes to generate theoretical ex post

sample paths that provide a better approximation to thesample paths of observed financial data

7 Conclusion

The classical ergodicity hypothesis provides a point of demar-cation in the prehistories of MPT and econophysics Todeal with the problem of making statistical inferences fromldquononexperimentalrdquo data theories in MPT typically employstationary densities that are time reversible are unimodaland allow no short or long term impact from initial andboundary conditionsThe possibility of bimodal processes orex ante impact from initial and boundary conditions is notrecognized or it seems intended Significantly as illustratedin the need to select an 119886

119894in Figure 1 in order to specify the

ldquoreal worldrdquo ex ante stationary density a semantic connectioncan be established between the subjective uncertainty aboutencountering a future bifurcation point and say the possiblecollapse of an asset price bubble impacting future marketvaluations Examining the quartic exponential stationarydistribution associatedwith a bifurcating ergodic process it isapparent that this distribution nests theGaussian distributionas a special case In this sense results from classical statisticalmechanics can be employed to produce a stochastic general-ization of the unimodal time reversible processes employedin modern portfolio theory

Appendix

Preliminaries on Solving theForward Equation

Due to the widespread application in a wide range of sub-jects textbook presentations of the Sturm-Liouville problempossess subtle differences that require some clarificationto be applicable to the formulation used in this paper Inparticular to derive the canonical form (8) of the Fokker-Planck equation (1) observe that evaluating the derivativesin (1) gives

119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)

This can be rewritten as

1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)

This provides the solution for the key function 119903[119909]

1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)

This 119903[119909] function is used to construct the scale and speeddensities commonly found in presentations of solutions tothe forward equation for example Karlin and Taylor [53] andLinetsky [49]

Another specification of the forward equation that is ofimportance is found in Wong [46 eq 6-7]

119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)

This formulation occurs after separating variables say with119880[119909] = 119892[119909]ℎ[119905] Substituting this result into (1) gives

120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)

Using the separation of variables substitution (1ℎ)120597ℎ120597119905 =minus120582 and redefining 119892[119909] = 120588120579 gives

119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)

Evaluating the derivative inside the bracket and using thecondition 119889119889119909[119861120588] minus 119860120588 = 0 to specify admissible 120588 give

119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)

which is equation (6) in Wong [46] The condition usedto define 120588 is then used to identify the specification of

119861[119909] and 119860[119909] from the Pearson system The associatedboundary condition follows from observing the 120588[119909] will bethe ergodic density andmaking appropriate substitutions intothe boundary condition

120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)

Evaluating the derivative and taking values at the lower (orupper) boundary give

119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)

Observing that the expression in the last bracket is theoriginal condition with the ergodic density serving as119880 givesthe boundary condition stated in Wong [46 eq 7]

Proof of Proposition 1 (a) 120595119899has exactly 119899 zeroes in [119886 119887]

Hille [51 p 398Theorem 833] and Birkhoff and Rota [52 p320 Theorem 5] show that the eigenfunctions of the Sturm-Liouville system (11015840) with (21015840) (31015840) and (4) have exactly 119899zeroes in the interval (119886 119887)More precisely since it is assumedthat 119903 gt 0 the eigenfunction 120595

119899corresponding to the 119899th

eigenvalue has exactly 119899 zeroes in (119886 119887)(b) For 120595

119899= 0 int119887119886120595

119899[119909]119889119909 = 0

Proof For 120595119899= 0 the following applies

120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)

Since each 120595119899[119909] satisfies the boundary conditions

(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)

But from part (b) this will = 0 (which is a contradiction)unless 120582

119896= 0 for some 119896

(d) Consider 1205820= 0

Proof From part (a) 1205950[119909] has no zeroes in (119886 119887) Therefore

either int1198871198861205950[119909]119889119909 gt 0 or int119887

1198861205950[119909]119889119909 lt 0

It follows from part (b) that 1205820= 0

(e) Consider 120582119899gt 0 for 119899 = 0 This follows from part (d)

and the strict inequality conditions provided in part (a)(f) Obtaining the solution for 119879[119909] in the Proposition

from part (d) it follows that

119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)

Integrating this equation from 119886 to 119909 and using the boundarycondition give

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)

This equation can be solved for 1205950to get

1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1

where 119862 = constant(A17)

Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)

Using this definition and observing that the integral of 119891[119909]over the state space is one it follows

1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)

(g) The Proof of the Proposition now follows from parts (f)(e) and (b)

Disclosure

The authors are Professor of Finance and Associate Professorof Finance at Simon Fraser University

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Helpful comments from Chris Veld Yulia Veld EmmanuelHavens Franck Jovanovic Marcel Boumans and ChristophSchinckus are gratefully acknowledged

Endnotes

1 For example Black and Scholes [4] discuss the con-sistency of the option pricing formula with the capi-tal asset pricing model More generally financial eco-nomics employs primarily Gaussian-based finite param-eter models that agree with the ldquotradeoff between riskand returnrdquo

2 In rational mechanics once the initial positions of theparticles of interest for example molecules are knownthe mechanical model fully determines the future evo-lution of the system This scientific and philosophicalapproach is often referred to as Laplacian determinism

3 Boltzmann and Max Planck were vociferous opponentsof energetics The debate over energetics was part of alarger intellectual debate concerning determinism andreversibility Jevons [66 p 738-9] reflects the entrencheddeterminist position of the marginalists ldquoWe may safelyaccept as a satisfactory scientific hypothesis the doctrineso grandly put forth by Laplace who asserted thata perfect knowledge of the universe as it existed atany given moment would give a perfect knowledgeof what was to happen thenceforth and for ever afterScientific inference is impossible unless we may regardthe present as the outcome of what is past and the causeof what is to come To the view of perfect intelligencenothing is uncertainrdquo What Boltzmann Planck andothers had observed in statistical physics was that eventhough the behavior of one or two molecules can becompletely determined it is not possible to generalizethesemechanics to the describe themacroscopicmotionof molecules in large complex systems for exampleBrush [67 esp ch II]

4 As such Boltzmann was part of the larger ldquoSecondScientific Revolution associated with the theories ofDarwin Maxwell Planck Einstein Heisenberg andSchrodinger (which) substituted a world of processand chance whose ultimate philosophical meaning stillremains obscurerdquo [67 p 79]This revolution supercededthe following ldquoFirst Scientific Revolution dominated bythe physical astronomy of Copernicus Kepler Galileoand Newton in which all changes are cyclic and allmotions are in principle determined by causal lawsrdquoThe irreversibility and indeterminism of the SecondScientific Revolution replaces the reversibility and deter-minism of the first


5 There are many interesting sources on these pointswhich provide citations for the historical papers that arebeing discussed Cercignani [68 p 146ndash50] discussesthe role of Maxwell and Boltzmann in the developmentof the ergodic hypothesis Maxwell [25] is identified asldquoperhaps the strongest statement in favour of the ergodichypothesisrdquo Brush [69] has a detailed account of thedevelopment of the ergodic hypothesis Gallavotti [70]traces the etymology of ldquoergodicrdquo to the ldquoergoderdquo in an1884 paper by Boltzmann More precisely an ergodeis shorthand for ldquoergomonoderdquo which is a ldquomonodewith given energyrdquo where a ldquomonoderdquo can be eithera single stationary distribution taken as an ensembleor a collection of such stationary distributions withsome defined parameterization The specific use is clearfrom the context Boltzmann proved that an ergodeis an equilibrium ensemble and as such provides amechanical model consistent with the second law ofthermodynamics It is generally recognized that themodern usage of ldquothe ergodic hypothesisrdquo originateswith Ehrenfest [71]

6 Kapetanios and Shin [37 p 620] capture the essence ofthis quandary ldquoInterest in the interface of nonstation-arity and nonlinearity has been increasing in the econo-metric literature The motivation for this developmentmay be traced to the perceived possibility that nonlinearergodic processes can be misinterpreted as unit rootnonstationary processes Furthermore the inability ofstandard unit root tests to reject the null hypothesis ofunit root for a large number of macroeconomic vari-ables which are supposed to be stationary according toeconomic theory is another reason behind the increasedinterestrdquo

7 The second law of thermodynamics is the universal lawof increasing entropy a measure of the randomness ofmolecular motion and the loss of energy to do workFirst recognized in the early 19th century the secondlaw maintains that the entropy of an isolated systemnot in equilibrium will necessarily tend to increaseover time Entropy approaches a maximum value atthermal equilibrium A number of attempts have beenmade to apply the entropy of information to problemsin economics with mixed success In addition to thesecond law physics now recognizes the zeroth law ofthermodynamics that ldquoany system approaches an equi-librium staterdquo [72 p 54] This implication of the secondlaw for theories in economics was initially explored byGeorgescu-Roegen [73]

8 In this process the ergodicity hypothesis is requiredto permit the one observed sample path to be used toestimate the parameters for the ex ante distribution ofthe ensemble paths In turn these parameters are usedto predict future values of the economic variable

9 Heterodox critiques are associated with views consid-ered to originate from within economics Such critiquesare seen to be made by ldquoeconomistsrdquo for example Post-Keynesian economists institutional economists radical

political economists and so on Because such critiquestake motivation from the theories of mainstream eco-nomics these critiques are distinct from econophysicsFollowing Schinckus [12 p 3818] ldquoEconophysicists havethen allies within economics with whom they shouldbecome acquaintedrdquo

10 Dhrymes [31 p 1ndash29] discusses the algebra of the lagoperator

11 Critiques of mainstream economics that are rooted inthe insights of The General Theory recognize the dis-tinction between fundamental uncertainty and objectiveprobability As a consequence the definition of ergodictheory in heterodox criticisms ofmainstream economicslacks formal precision for example the short termdependence of ergodic processes on initial conditions isnot usually recognized Ergodic theory is implicitly seenas another piece of the mathematical formalism inspiredby Hilbert and Bourbaki and captured in the Arrow-Debreu general equilibrium model of mainstream eco-nomics

12 In this context though not in all contexts econophysicsprovides a ldquomacroscopicrdquo approach In turn ergodicityis an assumption that permits the time average from asingle observed sample path to (phenomenologically)model the ensemble of sample paths Given this econo-physics does contain a substantively richer toolkit thatencompasses both ergodic and nonergodic processesMany works in econophysics implicitly assume ergod-icity and develop models based on that assumption

13 The distinction between invariant and ergodic measuresis fundamental Recognizing a number of distinct defi-nitions of ergodicity are available following Medio [74p 70] the Birkhoff-Khinchin ergodic (BK) theorem forinvariant measures can be used to demonstrate thatergodic measures are a class of invariant measuresMore precisely the BK theorem permits the limit ofthe time average to depend on initial conditions Ineffect the invariant measure is permitted to decomposeinto invariant ldquosubmeasuresrdquoThephysical interpretationof this restriction is that sample paths starting from aparticular initial condition may only be able to access apart of the sample space nomatter how long the processis allowed to run For an ergodic process sample pathsstarting fromany admissible initial conditionwill be ableto ldquofill the sample spacerdquo that is if the process is allowedto run long enough the time average will not depend onthe initial condition Medio [74 p 73] provides a usefulexample of an invariant measure that is not ergodic

14 The phenomenological approach is not without difficul-ties For example the restriction to Markov processesignores the possibility of invariant measures that arenot Markov In addition an important analytical con-struct in bifurcation theory the Lyapunov exponentcan encounter difficulties with certain invariant Markovmeasures Primary concern with the properties of thestationary distribution is not well suited to analysis of


the dynamic paths around a bifurcation point And soit goes

15 A diffusion process is ldquoregularrdquo if starting from anypoint in the state space 119868 any other point in 119868 can bereached with positive probability (Karlin and Taylor [53p 158] This condition is distinct from other definitionsof regular that will be introduced ldquoregular boundaryconditionsrdquo and ldquoregular S-L problemrdquo

16 The classification of boundary conditions is typically animportant issue in the study of solutions to the forwardequation Important types of boundaries include regu-lar exit entrance and natural Also the following areimportant in boundary classification the properties ofattainable and unattainable whether the boundary isattracting or non-attracting and whether the boundaryis reflecting or absorbing In the present context reg-ular attainable reflecting boundaries are usually beingconsidered with a few specific extensions to other typesof boundaries In general the specification of boundaryconditions is essential in determining whether a givenPDE is self-adjoint

17 Heuristically if the ergodic process runs long enoughthen the stationary distribution can be used to estimatethe constant mean value This definition of ergodicis appropriate for the one-dimensional diffusion casesconsidered in this paper Other combinations of trans-formation space and function will produce differentrequirements Various theoretical results are availablefor the case at hand For example the existence of aninvariant Markov measure and exponential decay of theautocorrelation function are both assured

18 For ease of notation it is assumed that 1199050

= 0 Inpractice solving (1) combined with (2)ndash(4) requires119886 and 119887 to be specified While 119886 and 119887 have readyinterpretations in physical applications for example theheat flow in an insulated bar determining these valuesin economic applications can bemore challenging Somesituations such as the determination of the distributionof an exchange rate subject to control bands (eg [75])are relatively straight forward Other situations such asprofit distributions with arbitrage boundaries or outputdistributions subject to production possibility frontiersmay require the basic S-L framework to be adapted tothe specifics of the modeling situation

19 The mathematics at this point are heuristic It would bemore appropriate to observe that 119880lowast is the special casewhere 119880 = Ψ[119909] a strictly stationary distribution Thiswould require discussion of how to specify the initial andboundary conditions to ensure that this is the solution tothe forward equation

20 A more detailed mathematical treatment can be foundin de Jong [76]

21 In what follows except where otherwise stated it isassumed that 120590 = 1 Hence the condition that 119870be a constant such that the density integrates to oneincorporates the 120590 = 1 assumption Allowing 120590 = 1 willscale either the value of 119870 or the 120573rsquos from that stated

22 Anumber of simplificationswere used to produce the 3Dimage in Figure 1 119909 has been centered about 120583 and 120590 =119870

119876= 1 Changing these values will impact the specific

size of the parameter values for a given 119909 but will notchange the general appearance of the density plots

References

[1] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol7 pp 77ndash91 1952

[2] J Y Campbell ldquoUnderstanding risk and returnrdquo Journal ofPolitical Economy vol 104 no 2 pp 298ndash345 1996

[3] P Mirowski ldquoPhysics and the lsquomarginalist revolutionrsquordquo Cam-bridge Journal of Economics vol 8 no 4 pp 361ndash379 1984

[4] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3 pp 637ndash659 1973

[5] B M Roehner Patterns of Speculation Cambridge UniversityPress Cambridge UK 2002

[6] F Jovanovic and C Schinckus ldquoEconophysics a new challengefor financial economicsrdquo Journal of the History of EconomicThought vol 35 no 3 pp 319ndash352 2013

[7] C Sprenkle ldquoWarrant prices as indicators of expectations andpreferencesrdquo Yale Economic Essays vol 1 pp 178ndash231 1961

[8] P Samuelson ldquoRational theory of Warrant pricingrdquo IndustrialManagement Review vol 6 pp 13ndash31 1965

[9] R C Merton ldquoThe theory of rational option pricingrdquo The BellJournal of Economics and Management Science vol 4 no 1 pp141ndash183 1973

[10] M J Brennan and E S Schwartz ldquoA continuous time approachto the pricing of bondsrdquo Journal of Banking and Finance vol 3no 2 pp 133ndash155 1979

[11] L G Epstein and S Ji ldquoAmbiguous volatility and asset pricingin continuous timerdquo Review of Financial Studies vol 26 no 7pp 1740ndash1786 2013

[12] C Schinckus ldquoIs econophysics a new discipline The neoposi-tivist argumentrdquo Physica A vol 389 no 18 pp 3814ndash3821 2010

[13] J L McCauleyDynamics of Markets Econophysics and FinanceCambridge University Press Cambridge UK 2004

[14] B B Mandelbrot Fractals and Scaling in Finance DiscontinuityConcentration Risk Springer New York NY USA 1997

[15] B B Mandelbrot and R L Hudson The (Mis)Behavior ofMarkets A Fractal View of Risk Ruin and Reward Basic BooksNew York NY USA 2004

[16] Ergodic Theory Encyclopedia of Mathematics Springer NewYork NY USA 2002

[17] J von Neumann ldquoA proof of the quasi-ergodic hypothesisrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 18 no 3 pp 255ndash263 1932

[18] G Birkhoff ldquoProof of the ergodic theoremrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 17 no 12 pp 656ndash660 1931

[19] J L Doob Stochastic Processes John Wiley amp Sons New YorkNY USA 1953

[20] GWMackey ldquoErgodic theory and its significance for statisticalmechanics and probability theoryrdquo Advances in Mathematicsvol 12 no 2 pp 178ndash268 1974

[21] A Kolmogorov Foundations of the Theory of ProbabilitySpringer Berlin Germany 1933 (German) English editionChelsea New York NY USA 1950


[22] L Arnold Random Dynamical Systems Springer Monographsin Mathematics Springer New York NY USA 1998

[23] P MirowskiMore HeatThan Light Economics as Social PhysicsPhysics as Naturersquos Economics Cambridge University PressCambridge UK 1989

[24] E R Weintraub How Economics Became a MathematicalScience Science and Cultural Theory Duke University PressDurham NC USA 2002

[25] J C Maxwell ldquoOn the dynamical theory of gasesrdquo PhilosophicalTransactions of the Royal Society vol 157 pp 49ndash88 1867

[26] NWeiner ldquoThe Ergodic theoremrdquoDukeMathematical Journalvol 5 no 1 pp 1ndash18 1939

[27] P A Samuelson ldquoOptimality of sluggish predictors underergodic probabilitiesrdquo International Economic Review vol 17no 1 pp 1ndash7 1976

[28] U Horst and J Wenzelburger ldquoOn non-ergodic asset pricesrdquoEconomic Theory vol 34 no 2 pp 207ndash234 2008

[29] A Dixit and R Pindyck Investment under Uncertainty Prince-ton University Press Princeton NJ USA 1994

[30] E Elton and M Gruber Modern Portfolio Theory and Invest-ment Analysis Wiley New York NY USA 2nd edition 1984

[31] P J Dhrymes Econometrics Statistical Foundations and Appli-cations Springer New York NY USA 1974

[32] HTheil Principles of Econometrics Wiley New York NY USA1971

[33] R F Engle and C W Granger ldquoCo-integration and error cor-rection representation estimation and testingrdquo Econometricavol 55 no 2 pp 251ndash276 1987

[34] M-C Beaulieu J-M Dufour and L Khalaf ldquoIdentification-robust estimation and testing of the zero-beta CAPMrdquo Reviewof Economic Studies vol 80 no 3 pp 892ndash924 2013

[35] D F Hendry Dynamic Econometrics Oxford University PressOxford UK 1995

[36] D A Dickey and W A Fuller ldquoDistribution of the estimatorsfor autoregressive time series with a unit rootrdquo Journal of theAmerican Statistical Association vol 74 pp 427ndash431 1979

[37] G Kapetanios and Y Shin ldquoTesting the null hypothesis ofnonstationary long memory against the alternative hypothesisof a nonlinear ergodic modelrdquo Econometric Reviews vol 30 no6 pp 620ndash645 2011

[38] M Bonomo R Garcia N Meddahi and R Tedongap ldquoGen-eralized disappointment aversion long-run volatility risk andasset pricesrdquo Review of Financial Studies vol 24 no 1 pp 82ndash122 2011

[39] P R Halmos ldquoMeasurable transformationsrdquo Bulletin of theAmerican Mathematical Society vol 55 pp 1015ndash1034 1949

[40] P Davidson ldquoIs probability theory relevant for uncertainty Apost Keynesian perspectiverdquo Journal of Economic Perspectivesvol 5 no 1 pp 129ndash143 1991

[41] E Dimson P Marsh and M Staunton Triumph of theOptimists 101 Years of Global Investment Returns PrincetonUniversity Press Princeton NJ USA 2002

[42] G M Constantinides ldquoRational asset pricesrdquo Journal ofFinance vol 57 no 4 pp 1567ndash1591 2002

[43] C Chiarella X-Z He DWang andM Zheng ldquoThe stochasticbifurcation behaviour of speculative financial marketsrdquo PhysicaA Statistical Mechanics and Its Applications vol 387 no 15 pp3837ndash3846 2008

[44] H Crauel P Imkeller and M Steinkamp ldquoBifurcations ofone-dimensional stochastic differential equationsrdquo in Stochastic

Dynamics H Crauel and M Gundlach Eds chapter 2 pp 27ndash47 Springer New York NY USA 1999

[45] W Feller ldquoDiffusion processes in one dimensionrdquo Transactionsof the American Mathematical Society vol 77 pp 1ndash31 1954

[46] E Wong ldquoThe construction of a class of stationary Markoffprocessesrdquo in Proceedings of the 16th Symposium on AppliedMathematics R Bellman Ed American Mathmatical SocietyProvidence RI USA 1964

[47] I Gihman and A SkorohodTheTheorv of Stochastic Processesvol 1ndash3 Springer New York NY USA 1979

[48] H RiskenTheFokker-Planck EquationMethods of Solution andApplications vol 18 of Springer Series in Synergetics SpringerNew York NY USA 1989

[49] V Linetsky ldquoOn the transition densities for reflected diffusionsrdquoAdvances inApplied Probability vol 37 no 2 pp 435ndash460 2005

[50] P W Berg and J L McGregor Elementary Partial DifferentialEquations Holden-Day San Francisco Calif USA 1966

[51] E Hille Lectures on Ordinary Differential Equations Addison-Wesley London UK 1969

[52] G Birkhoff and G-C Rota Ordinary Differential EquationsJohn Wiley amp Sons New York NY USA 4th edition 1989

[53] S Karlin and H M Taylor A Second Course in StochasticProcesses Academic Press New York NY USA 1981

[54] D R Cox and H D Miller The Theory of Stochastic ProcessesChapman amp Hall London UK 1965

[55] L PHansen J A Scheinkman andN Touzi ldquoSpectralmethodsfor identifying scalar diffusionsrdquo Journal of Econometrics vol86 no 1 pp 1ndash32 1998

[56] Y Aıt-Sahalia ldquoTransition densities for interest rates and othernonlinear diffusionsrdquo Journal of Finance vol 54 pp 1361ndash13951999

[57] D Veerstraeten ldquoThe conditional probability density functionfor a reflected Brownian motionrdquo Computational Economicsvol 24 no 2 pp 185ndash207 2004

[58] L Cobb ldquoStochastic catastrophemodels andmultimodal distri-butionsrdquo Behavioral Science vol 23 no 5 pp 360ndash374 1978

[59] L Cobb ldquoThe multimodal exponential families of statisticalcatastrophe theoryrdquo in Statistical Distributions in ScientificWork Vol 4 (Trieste 1980) C Taillie G Patil and B BaldessariEds vol 79 of NATO Adv Study Inst Ser C Math Phys Scipp 67ndash90 Reidel Dordrecht The Netherlands 1981

[60] H Crauel and F Flandoli ldquoAdditive noise destroys a pitchforkbifurcationrdquo Journal of Dynamics and Differential Equationsvol 10 no 2 pp 259ndash274 1998

[61] R A Fisher ldquoOn the mathematical foundations of theoreticalstatisticsrdquo Philosophical Transactions of the Royal Society A vol222 pp 309ndash368 1921

[62] L Cobb P Koppstein and N H Chen ldquoEstimation andmoment recursion relations for multimodal distributions ofthe exponential familyrdquo Journal of the American StatisticalAssociation vol 78 no 381 pp 124ndash130 1983

[63] J Elliott ldquoEigenfunction expansions associated with singulardifferential operatorsrdquo Transactions of the AmericanMathemat-ical Society vol 78 pp 406ndash425 1955

[64] W Horsthemke and R Lefever Noise-Induced Transitions vol15 of Springer Series in Synergetics Springer Berlin Germany1984

[65] AWMatz ldquoMaximum likelihood parameter estimation for thequartic exponential distributionrdquo Technometrics vol 20 no 4pp 475ndash484 1978


[66] W S Jevons The Principles of Science Dover Press New YorkNY USA 2nd edition 1877 (Reprint 1958)

[67] S Brush Statistical Physics and the Atomic Theory of MatterFrom Boyle and Newton to Landau and Onsager PrincetonUniversity Press Princeton NJ USA 1983

[68] C Cercignani Ludwig BoltzmannTheManwho Trusted AtomsOxford University Press Oxford UK 1998

[69] S Brush The Kind of Motion We Call Heat A History of theKinetic Theory of Gases in the 19th Century North HollandAmsterdam The Netherlands 1976

[70] G Gallavotti ldquoErgodicity ensembles irreversibility in Boltz-mann and beyondrdquo Journal of Statistical Physics vol 78 no 5-6pp 1571ndash1589 1995

[71] P EhrenfestBegriffliche Grundlagen der statistischenAuffassungin der Mechanik BG Teubner Leipzig Germany 1911

[72] M Reed and B Simon Methods of Modern MathematicalPhysics Academic Press NewYork NYUSA 2nd edition 1980

[73] N Georgescu-Roegen The Entropy Law and the EconomicProcess HarvardUniversity Press CambridgeMass USA 1971

[74] AMedio ldquoErgodic theory of nonlinear dynamicsrdquo inNonlinearDynamical Systems in Economics M Lines Ed vol 476 ofCISM International Centre for Mechanical Sciences pp 67ndash101Springer Vienna Austria 2005

[75] C A Ball and A Roma ldquoDetecting mean reversion withinreflecting barriers application to the European Exchange RateMechanismrdquo Applied Mathematical Finance vol 5 no 1 pp 1ndash15 1998

[76] F de Jong ldquoA univariate analysis of European monetary systemexchange rates using a target zone modelrdquo Journal of AppliedEconometrics vol 9 no 1 pp 31ndash45 1994

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


pricing model and other elements of MPT1 Significantlystochastic generalization of the static equilibrium approachof MPT required the adoption of a restricted class of ergodicprocesses that is ldquotime reversiblerdquo probabilistic modelsespecially the unimodal likelihood functions associated withcertain stationary distributions

In contrast from the early ergodic models of Boltzmannto the fractals and chaos theory of Mandlebrot physicshas employed a wider variety of ergodic and nonergodicstochastic models aimed at capturing key empirical charac-teristics of various physical problems at hand Such modelstypically have a mathematical structure that varies fromthe constrained optimization techniques underpinningMPTrestricting the straightforward application of many physicalmodels Yet the demarcation between the use of ergodicnotions in physics and financial economics was blurred sub-stantively by the introduction of diffusion process techniquesto solve contingent claims valuation problems Followingcontributions by Sprenkle [7] and Samuelson [8] Blackand Scholes [4] and Merton [9] provided an empiricallyviable method of using diffusion methods to determineusing Itorsquos lemma a partial differential equation that canbe solved for an option price Use of Itorsquos lemma to solvestochastic optimization problems is now commonplace infinancial economics for example Brennan and Schwartz [10]including the continuous time generalizations of MPT forexample Epstein and Ji [11] In spite of the considerableprogression of certain mathematical techniques employed inphysics into financial economics overcoming the difficultiesof applying the wide range of models developed for physicalsituations to fit the empirical properties of financial data isstill a central problem confronting econophysics

Schinckus [12 p 3816] accurately recognizes that thepositivist philosophical foundation of econophysics dependsfundamentally on empirical observation ldquoThe empiricistdimension is probably the first positivist feature of econo-physicsrdquo Following McCauley [13] and others this con-cern with empiricism often focuses on the identification ofmacrolevel statistical regularities that are characterized bythe scaling laws identified by Mandelbrot [14] and Man-dlebrot and Hudson [15] for financial data Unfortunatelythis empirically driven ideal is often confounded by theldquononrepeatablerdquo experiment that characterizesmost observedeconomic and financial data There is quandary posed byhaving only a single observed ex post time path to estimatethe distributional parameters for the ensemble of ex antetime paths needed to make decisions involving future valuesof financial variables In contrast to natural sciences suchas physics in the human sciences there is no assurancethat ex post statistical regularity translates into ex ante fore-casting accuracy Resolution of this quandary highlights theusefulness of employing a ldquophenomenologicalrdquo approach tomodeling stochastic properties of financial variables relevantto MPT

To this end this paper provides an etymology andhistory of the ldquoclassical ergodicity hypothesisrdquo in 19th cen-tury statistical mechanics Subsequent use of ergodicity infinancial economics in general and MPT in particularis then examined A modern interpretation of classical

ergodicity is provided that uses Sturm-Liouville theory amathematical method central to classical statistical mechan-ics to decompose the transition probability density of aone-dimensional diffusion process subject to regular upperand lower reflecting barriers This ldquoclassicalrdquo decompositiondivides the transition density of an ergodic process intoa possibly multimodal limiting stationary density whichis independent of time and initial condition and a powerseries of time and boundary dependent transient terms Incontrast empirical theory aimed at estimating relationshipsfrom MPT typically ignores the implications of the initialand boundary conditions that generate transient terms andfocuses on properties of a particular class of unimodal lim-iting stationary densities with finite parameters To illustratethe implications of the expanded class of ergodic processesavailable to econophysics properties of the bimodal quarticexponential stationary density are considered and used toassess the ability of the classical ergodicity hypothesis toexplain certain ldquostylized factsrdquo associated with the ex postexante quandary confronting MPT

2 A Brief History of Classical Ergodicity

The Encyclopedia of Mathematics [16] defines ergodic theoryas the ldquometric theory of dynamical systems the branch ofthe theory of dynamical systems that studies systems withan invariant measure and related problemsrdquo This moderndefinition implicitly identifies the birth of ergodic theorywith proofs of the mean ergodic theorem by von Neumann[17] and the pointwise ergodic theorem by Birkhoff [18]These early proofs have had significant impact in a widerange of modern subjects For example the notions ofinvariant measure and metric transitivity used in the proofsare fundamental to the measure theoretic foundation ofmodern probability theory [19 20] Building on a seminalcontribution to probability theory by Kolmogorov [21] inthe years immediately following it was recognized that theergodic theorems generalize the strong law of large numbersSimilarly the equality of ensemble and time averagesmdashtheessence of the mean ergodic theoremmdashis necessary to theconcept of a strictly stationary stochastic process Ergodictheory is the basis for themodern study of randomdynamicalsystems for example Arnold [22] In mathematics ergodictheory connects measure theory with the theory of transfor-mation groups This connection is important in motivatingthe generalization of harmonic analysis from the real line tolocally compact groups

From the perspective of modern mathematics statisticalphysics or systems theory Birkhoff [18] and von Neumann[17] are excellent starting points for a modern history ofergodic theory Building on the modern ergodic theoremssubsequent developments in these and related fields havebeen dramatic These contributions mark the solution to aproblem in statistical mechanics and thermodynamics thatwas recognized sixty years earlier when Ludwig Boltzmannintroduced the classical ergodic hypothesis to permit the the-oretical phase space average to be interchanged with themea-surable time average For the purpose of contrastingmethods





















Ψ[119871


119896)] and (119871

lowast)


















119900= (119886 119887




120597

2

120597119909

2 119861 [119909]119880 minus120597

120597119909

119860 [119909]119880 =

120597119880

120597119905

(1)




120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886]119880 [119886 119905] = 0 (2)

120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887]119880 [119887 119905] = 0 (3)


119880 [119909 0] = 119891 [1199090] where int119887

119886

119891 [1199090] = 1 (4)



0




lim119905rarrinfin

119880 [119909 119905 | 1199090] = int119887

119886

119891 [1199090] 119880 [119909 119905 | 1199090] 1198891199090

= Ψ [119909]

(5)


0]) and when 119891[119909




018 Time






0as an





119889Ψ [119909]

119889119909

=

1198901119909 + 11989001198892119909

2+ 1198891119909 + 1198890

Ψ [119909] (6)


0 1198891 1198892 1198900 and 119890

1in the



119903 [119909] = 119861 [119909] exp [minusint119909

119886

119860 [119904]

119861 [119904]

119889119904] (7)


1119903 [119909]

120597

120597119909

119901 [119909]

120597119880

120597119909

+ 119902 [119909]119880 =

120597119880

120597119905

(8)

where119901 [119909] = 119861 [119909] 119903 [119909]

119902 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(9)



119880 [119909 119905] = 119890

minus120582119905120593 [119909]

(10)


1119903 [119909]

119889

119889119909

[119901 [119909]

119889120593

119889119909

]+ [119902 [119909] + 120582] 120593 [119909] = 0 (11015840)



119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)




0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)


119899equiv 120593


120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)



2[119886 119887] to (1) subject to


119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)


0= 0



119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)



119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)




119868



119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)


2

21205902119905] (19)








119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Ψ[119871


119896)] and (119871

lowast)


















119900= (119886 119887




120597

2

120597119909

2 119861 [119909]119880 minus120597

120597119909

119860 [119909]119880 =

120597119880

120597119905

(1)




120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886]119880 [119886 119905] = 0 (2)

120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887]119880 [119887 119905] = 0 (3)


119880 [119909 0] = 119891 [1199090] where int119887

119886

119891 [1199090] = 1 (4)



0




lim119905rarrinfin

119880 [119909 119905 | 1199090] = int119887

119886

119891 [1199090] 119880 [119909 119905 | 1199090] 1198891199090

= Ψ [119909]

(5)


0]) and when 119891[119909




018 Time






0as an





119889Ψ [119909]

119889119909

=

1198901119909 + 11989001198892119909

2+ 1198891119909 + 1198890

Ψ [119909] (6)


0 1198891 1198892 1198900 and 119890

1in the



119903 [119909] = 119861 [119909] exp [minusint119909

119886

119860 [119904]

119861 [119904]

119889119904] (7)


1119903 [119909]

120597

120597119909

119901 [119909]

120597119880

120597119909

+ 119902 [119909]119880 =

120597119880

120597119905

(8)

where119901 [119909] = 119861 [119909] 119903 [119909]

119902 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(9)



119880 [119909 119905] = 119890

minus120582119905120593 [119909]

(10)


1119903 [119909]

119889

119889119909

[119901 [119909]

119889120593

119889119909

]+ [119902 [119909] + 120582] 120593 [119909] = 0 (11015840)



119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)




0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)


119899equiv 120593


120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)



2[119886 119887] to (1) subject to


119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)


0= 0



119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)



119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)




119868



119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)


2

21205902119905] (19)








119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119900= (119886 119887




120597

2

120597119909

2 119861 [119909]119880 minus120597

120597119909

119860 [119909]119880 =

120597119880

120597119905

(1)




120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886]119880 [119886 119905] = 0 (2)

120597

120597119909

119861 [119909]119880 [119909 119905]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887]119880 [119887 119905] = 0 (3)


119880 [119909 0] = 119891 [1199090] where int119887

119886

119891 [1199090] = 1 (4)



0




lim119905rarrinfin

119880 [119909 119905 | 1199090] = int119887

119886

119891 [1199090] 119880 [119909 119905 | 1199090] 1198891199090

= Ψ [119909]

(5)


0]) and when 119891[119909




018 Time






0as an





119889Ψ [119909]

119889119909

=

1198901119909 + 11989001198892119909

2+ 1198891119909 + 1198890

Ψ [119909] (6)


0 1198891 1198892 1198900 and 119890

1in the



119903 [119909] = 119861 [119909] exp [minusint119909

119886

119860 [119904]

119861 [119904]

119889119904] (7)


1119903 [119909]

120597

120597119909

119901 [119909]

120597119880

120597119909

+ 119902 [119909]119880 =

120597119880

120597119905

(8)

where119901 [119909] = 119861 [119909] 119903 [119909]

119902 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(9)



119880 [119909 119905] = 119890

minus120582119905120593 [119909]

(10)


1119903 [119909]

119889

119889119909

[119901 [119909]

119889120593

119889119909

]+ [119902 [119909] + 120582] 120593 [119909] = 0 (11015840)



119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)




0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)


119899equiv 120593


120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)



2[119886 119887] to (1) subject to


119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)


0= 0



119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)



119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)




119868



119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)


2

21205902119905] (19)








119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0




lim119905rarrinfin

119880 [119909 119905 | 1199090] = int119887

119886

119891 [1199090] 119880 [119909 119905 | 1199090] 1198891199090

= Ψ [119909]

(5)


0]) and when 119891[119909




018 Time






0as an





119889Ψ [119909]

119889119909

=

1198901119909 + 11989001198892119909

2+ 1198891119909 + 1198890

Ψ [119909] (6)


0 1198891 1198892 1198900 and 119890

1in the



119903 [119909] = 119861 [119909] exp [minusint119909

119886

119860 [119904]

119861 [119904]

119889119904] (7)


1119903 [119909]

120597

120597119909

119901 [119909]

120597119880

120597119909

+ 119902 [119909]119880 =

120597119880

120597119905

(8)

where119901 [119909] = 119861 [119909] 119903 [119909]

119902 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(9)



119880 [119909 119905] = 119890

minus120582119905120593 [119909]

(10)


1119903 [119909]

119889

119889119909

[119901 [119909]

119889120593

119889119909

]+ [119902 [119909] + 120582] 120593 [119909] = 0 (11015840)



119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)




0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)


119899equiv 120593


120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)



2[119886 119887] to (1) subject to


119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)


0= 0



119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)



119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)




119868



119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)


2

21205902119905] (19)








119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

minus119860 [119886] 120593 [119886] = 0 (21015840)

119889

119889119909

119861 [119909] 120593 [119909]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120593 [119887] = 0 (31015840)




0lt 120582

1lt 120582

2lt sdot sdot sdot

with

lim119899rarrinfin

120582

119899= infin (11)


119899equiv 120593


120595

119899 [119909] = [int

119887

119886

119903 [119909] 120593119899

2119889119909]

minus12

120593

119899

(12)



2[119886 119887] to (1) subject to


119880 [119909 119905] =

infin

sum

119899=0119888

119899120595

119899 [119909] 119890

minus120582119899119905 (13)

where

119888

119899= int

119887

119886

119903 [119909] 119891 [1199090] 120595119899 [119909] 119889119909 (14)


0= 0



119880 [119909 119905 | 1199090] = Ψ [119909] +119879 [119909 119905 | 1199090] (15)

where

Ψ [119909] =

119903 [119909]

minus1

int

119887

119886119903 [119909]

minus1119889119909

(16)



119879[119909 119905] are defined as

119879 [119909 119905 | 1199090] =infin

sum

119899=1119888

119899119890

minus120582119899119905120595

119899 [119909]

=

1119903 [119909]

infin

sum

119899=1119890

minus120582119899119905120595

119899 [119909] 120595119899

[1199090]

(17)

with

int

119887

119886

119879 [119909 119905 | 1199090] 119889119909 = 0

lim119905rarrinfin

119879 [119909 119905 | 1199090] = 0(18)




119868



119880 [119909 119905 | 1199090] =1

120590radic(2120587119905)


2

21205902119905] (19)








119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119880 [119909 119905 | 1199090] =1

120590radic2120587119905


2

21205902119905)


2

21205902119905)

+

1120590radic2120587119905

2120583120590

2

sdot exp(2120583119909120590

2 )(1minus119873[

119909 + 1199090 + 120583119905

120590radic119905

])

(20)



Ψ [119909] =

2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

120590

2 expminus(2 1003816100381610038161003816

120583

1003816

1003816

1003816

1003816

119909

120590

2 ) (21)



0] minus Ψ[119909]



Ψ [119909] =

1radic2120587

exp[minus1199092

2]

119868


(22)


120597

2119880

120597119909

2 +120597

120597119909

119909119880 =

120597119880

120597119905

(23)


119880 [119909 119905 | 1199090]

=

1



2

2 (1 minus 119890minus2119905)]

(24)




119889119909

119889119905

= minus 119909

3+1205881119909+ 1205880 (25)

where 1205880and 120588
















119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119889119883 (119905) = (1205881119883 (119905) minus119883 (119905)

3) 119889119905 + 120590119889119882 (119905) (26)




Ψ [119909] = 119870 exp [minusΦ [119909]]

= 119870 exp [minus (12057341199094+1205733119909

3+1205732119909

2+1205731119909)]

(27)




4= 120573

3=

120573

1= 0 and 120573

2= 12 with 119870 = 1(



Ψ [119910] = 119870

119878exp [minus 1205732 (119909 minus 120583)

2+1205734 (119909 minus 120583)

4]

where 1205734 ge 0(28)


1= 120573



34120573

4 to


Ψ [119910] = 119870

119876exp [minus 120581 (119910 minus 120583

119910) + 120572 (119910 minus 120583

119910)

2

+ 120574 (119910 minus 120583

119910)

4] where 120574 ge 0

(29)


120581 =

8120573112057342minus 4120573212057331205734 + 1205733

3

812057342

120572 =

812057321205734 minus 312057332

81205734

120574 = 1205734

(30)


3and 120573




ary density Ψ119894[119909] = 119870


119894119909) defines





120572 = 120573


119886







0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2x

0

05

1

a

0

04030201

minus2

minus1

minus05

Ψi[x]


119894[119909] =

119870





|119886



119894 In this the







0



7 Conclusion




Appendix



119861 [119909]

120597

2119880

120597119909

2 +[2120597119861

120597119909

minus119860 [119909]]

120597119880

120597119909

+[

120597

2119861

120597119909

2 minus120597119860

120597119909

]119880

=

120597119880

120597119905

(A1)


1119903 [119909]

120597

120597119909

[119875 [119909]

120597119880

120597119909

]+119876 [119909]119880 =

120597119880

120597119905

(A2)

where

119875 [119909] = 119861 [119909] 119903 [119909]

1119903 [119909]

120597119875

120597119909

= 2120597119861120597119909

minus119860 [119909]

119876 [119909] =

120597

2119861

120597119909

2 minus120597119860

120597119909

(A3)


It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










It follows that

120597119861

120597119909

=

1119903 [119909]

120597119875

120597119909

minus

1119903

2120597119903

120597119909

119875 [119909]

= 2120597119861120597119909

minus119860 [119909] minus

119861 [119909]

119903 [119909]

120597119903

120597119909

(A4)


1119903 [119909]

120597119903

120597119909

minus

1119861 [119909]

120597119861

120597119909

= minus

119860 [119909]

119861 [119909]

997888rarr ln [119903] minus ln [119896]

= minusint

119909119860 [119904]

119861 [119904]

119889119904

119903 [119909] = 119861 [119909] exp [minusint119909119860 [119904]

119861 [119904]

119889119904]

(A5)



119889

119889119909

[119861 [119909] 120588 [119909]

119889120579

119889119909

]+120582120588 [119909] 120579 [119909] = 0

with bc 119861 [119909] 120588 [119909] 119889120579119889119909

= 0(A6)


120597

2

120597119909

2 [119861119892ℎ] minus120597

120597119909

[119860119892ℎ] = 119892 [119909]

120597ℎ

120597119905

(A7)


119889

119889119909

[

119889

119889119909

119861119892minus119860119892]

= minus120582119892

=

119889

119889119909

[

119889

119889119909

119861 [119909] 120588 [119909] 120579 [119909] minus119860 [119909] 120588 [119909] 120579 [119909]]

= minus 120582120588120579

(A8)


119889

119889119909

[120579

119889

119889119909

(119861120588) +119861120588

119889

119889119909

120579 minus119860120588120579]

=

119889

119889119909

[119861120588

119889

119889119909

120579] = minus120582120588 [119909] 120579 [119909]

(A9)



120597

120597119909

119861 [119909] 119891 [119905] 120588 [119909] 120579 [119909] minus119860 [119909] 119891 [119905] 120588 [119909] 120579 [119909]

= 0

997888rarr

119889

119889119909

[119861120588120579] minus119860120588120579 = 0

(A10)


119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886]

119889119861120588

119889119909

minus119860 [119886] 120588 [119886] 120579 [119886]

= 0

= 119861 [119886] 120588 [119886]

119889120579 [119886]

119889119909

+ 120579 [119886] [

119889119861 [119886] 120588 [119886]

119889119909

minus119860 [119886] 120588 [119886]]

(A11)






119899= 0 int119887119886120595

119899[119909]119889119909 = 0


120595

119899=

1120582

119899

119889

119889119909

119889

119889119909

[119861 [119909] 120595119899] minus119860 [119909] 120595119899

there4 int

119887

119886

120595

119899 [119909] 119889119909 =

1120582

119899

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119887

minus119860 [119887] 120595119899 [119887] minus

119889

119889119909

[119861 [119909] 120595119899]

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119886

+119861 [119886] 120595119899 [119886] = 0

(A12)


(c) For some 119896 120582119896= 0

Proof From (13)

119880 [119909 119905] =

infin

sum

119896=0119888

119896119890

minus120582119896119905120595

119896 [119909] (A13)


Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since int119887119886119880[119909 119905]119889119909 = 1 then

1 =infin

sum

119896=0119888

119896119890

minus120582119896119905int

119887

119886

120595

119896 [119909] 119889119909 (A14)


119896= 0 for some 119896

(d) Consider 1205820= 0



1198861205950[119909]119889119909 lt 0





119889

119889119909

[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A15)


[119861 [119909] 1205950 [119909 119905]]119909 minus119860 [119909] 1205950 [119909 119905] = 0 (A16)


1205950 = 119860 [119861 [119909]]minus1 exp [int

119909

119886

119860 [119904]

119861 [119904]

119889119904] = 119862 [119903 [119909]]

minus1


Therefore

1205950 [119909] = [int119887

119886

119903 [119909] 119862

2[119903 [119909]]

minus2119889119909]

minus12

119862 [119903 [119909]]

minus1

=

[119903 [119909]]

minus1

[int

119887

119886119903 [119909]

minus1119889119909]

12

(A18)


1198880 = int119887

119886

119891 [119909] 119903 [119909]

119903 [119909]

minus1

[int

119887

119886119903 [119909] 119889119909]

12

119889119909

=

1

[int

119887

119886119903 [119909] 119889119909]

12

there4 1198880120595 [119909] =119903 [119909]

minus1

[int

119887

119886[119903 [119909]]

minus1119889119909]

(A19)


Disclosure




Acknowledgments


Endnotes






























References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































References























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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