Physica A 391 (2012) 5598–5610

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Physica A

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A Langevin approach to the Log–Gauss–Pareto compositestatistical structureIddo I. Eliazar a,∗, Morrel H. Cohen b,c

a Holon Institute of Technology, P.O. Box 305, Holon 58102, Israelb Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019, USAc Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

a r t i c l e i n f o

Article history:Received 16 April 2012Received in revised form 2 June 2012Available online 23 June 2012

Keywords:Ornstein–Uhlenbeck dynamicsLinear forcesGauss distributionLangevin dynamicsSigmoidal forcesExponential tailsGeometric Ornstein–Uhlenbeck dynamicsLog–Gauss distributionGeometric Langevin dynamicsPower-law tailsLaplace distributionLog–Laplace distribution‘‘mild’’ randomness‘‘wild’’ randomnessUniversality

a b s t r a c t

The distribution of wealth in human populations displays a Log–Gauss–Pareto compositestatistical structure: its density is Log–Gauss in its central body, and follows power-law decay in its tails. This composite statistical structure is further observed in othercomplex systems, and on a logarithmic scale it displays a Gauss-Exponential structure:its density is Gauss in its central body, and follows exponential decay in its tails. In thispaper we establish an equilibrium Langevin explanation for this statistical phenomenon,and show that: (i) the stationary distributions of Langevin dynamics with sigmoidal forcefunctions display a Gauss-Exponential composite statistical structure; (ii) the stationarydistributions of geometric Langevin dynamics with sigmoidal force functions display aLog–Gauss–Pareto composite statistical structure. This equilibrium Langevin explanationis universal — as it is invariant with respect to the specific details of the sigmoidalforce functions applied, and as it is invariant with respect to the specific statistics of theunderlying noise.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

At the close of the nineteenth century, after studying reams of income and tax data, the Italian economist Vilfredo Paretocame up with a remarkable empirical discovery regarding the distribution of wealth in human populations [1]. Paretodiscovered that the frequency of individuals with wealth greater than a given value v follows, asymptotically, a decreasingpower-law in the variable v. The striking feature of Pareto’s discovery was its empirical prevalence—all human populationsstudied by Pareto appeared to be governed by the aforementioned asymptotic power-law statistics, albeit with differentexponents. Contemporary economics corroborate Pareto’s findings and further assert that [2]: The distribution of wealth inhuman populations is Log-Gauss (i.e., log-normal) in its central body, and is Pareto (i.e., power-law) in its upper tail (v → ∞).

The aforementioned Log-Gauss–Pareto composite statistical structure is observed in various fields of science, examplesincluding (see Ref. [3] and references therein): income distributions, sizes of human settlements, sizes of sand and diamondparticles, sizes of oil fields, returns of stock prices, sizes of WWW sites, and sizes of computer files. Moreover, in all these

∗ Corresponding author. Tel.: +972 507 290 650.E-mail addresses: eliazar@post.tau.ac.il (I.I. Eliazar), mcohen@physics.rutgers.edu (M.H. Cohen).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.06.024

I.I. Eliazar, M.H. Cohen / Physica A 391 (2012) 5598–5610 5599

examples the asymptotic power-law behavior holds in the lower tail (v → 0), as well as in the upper tail (v → ∞). TheLog-Gauss–Pareto composite structure also arises in insurance as an actuarial model for loss distributions [4–6].

In the scientific literature there is a host of theoretical models for the generation of both Log-Gauss and Paretodistributions [7–9]. However, there are very few theoretical models for the generation of the Log-Gauss–Pareto compositestructure. The most straightforward approach to obtain the Log-Gauss–Pareto composite structure is to simply stitchtogether the Log-Gauss and Pareto distributions—the former up to a threshold level, and the latter above the thresholdlevel [4–6]. This approach, while empirically efficient and applicable, is not explanatory. A generative model for the Log-Gauss–Pareto composite structure, providing an explanation for its emergence in the context of file sizes, is the RecursiveForest File model [10].

A more general generative model for the Log-Gauss–Pareto composite structure is the ‘‘Double-Pareto Log-Normal’’ (DPLN) model [3,11]. The DPLN model is based on the phenomena that prevalent stochastic multiplicative growthprocesses, when stopped at random exponential times, yield asymptotic power-law statistics [12]. The quintessentialexample of stochastic multiplicative growth processes is Geometric Brownian Motion (GBM)—which is commonly appliedin economics and finance [13,14], and which underlies the Merton–Black–Scholes option pricing formula [15,16]. The DPLNmodel attains the Log-Gauss–Pareto composite structure by running GBM for a fixed deterministic time, and thereafter foran additional random exponential time.

In this paper we establish a diametrically different theoretical model for the generation of the Log-Gauss–Paretocomposite structure. While the DPLN model is based on the random stopping of the non-equilibrium GBM, the model wepropose herein is based on the equilibria of geometric Langevin dynamics.

Langevin dynamics are one of the most elemental forms of stochastic dynamics in the physical sciences [17,18]. TheLangevin equation describes the dynamics of a motion which is driven by a deterministic force towards a fixed point, whilesimultaneously being perturbed off the fixed point by a random white noise. The result of these antithetical effects – thedeterministic force and the randomwhite noise – is a stochastic equilibrium level towhich the Langevin dynamics converge.

The Langevin dynamics are additive—as they measure the motion’s change in terms of the motion’s displacement. Inmany dynamical settings however – economic and financial settings [13,14], as well as physical settings [19–21] – thedynamical change of themotion ismeasured in terms of themotion’s yields. In such settings the additive Langevin dynamicsare replaced by multiplicative geometric Langevin dynamics—which are the exponentiation of Langevin dynamics, and alsoconverge to a stochastic equilibrium level.

In this paper we show that geometric Langevin dynamics universally generate a Log-Gauss–Pareto composite structure.Indeed, for a ‘‘universality class’’ of geometric Langevin dynamics the probability density function of the stochasticequilibrium level is shown todisplay: (i) a Log-Gauss structure in its central body; (ii) asymptotic power-law statistics both inits lower tail (v → 0) and in its upper tail (v → ∞). The aforementioned probability density function admits two universalshapes—a monotone decreasing shape, and a unimodal shape. We further show that it is the magnitude of the randomwhite noise that determines which of the two universal shapes—monotone decreasing, or unimodal—shall be expressed bythe geometric Langevin dynamics. The monotone decreasing shape is encountered in the context of first passage times [22]in anomalous diffusion [23–25]—inwhich case it is theoretically explained by the non-equilibrium continuous time randomwalk model [26,27]. The unimodal shape is encountered in the abovementioned examples—income distributions, sizes ofhuman settlements, sizes of sand and diamond particles, sizes of oil fields, returns of stock prices, sizes of WWW sites, andsizes of computer files (see Ref. [3] and references therein).

Thus, in this paper we establish an equilibrium model – based on geometric Langevin dynamics – for the universalgeneration the Log-Gauss–Pareto composite statistical structure. The organization of the paper is as follows. We begin withOrnstein–Uhlenbeck dynamics (Section 2), shift to Langevin dynamics (Section 3), and then shift to geometric Langevindynamics (Section 4). In Section 3 we demonstrate how a Gauss-Exponential composite structure emerges from Langevindynamics, and in Section 4we demonstrate how a Log-Gauss–Pareto composite structure emerges from geometric Langevindynamics. These general results are followed up by a concrete example (Section 5), a stochastic-limit analysis (Section 6),a discussion of the ‘‘tails thickening’’ phenomenon of the aforementioned composite statistical structures (Section 7), anapplication to financial markets (Section 8), and a comparison to the DPLN model (Section 9). The universality of theequilibrium model is discussed in Section 10.

2. Linear forces and Ornstein–Uhlenbeck dynamics

The linear ordinary differential equation (ODE)

X(t) = −r (X(t)− l) , (1)

where r is a positive parameter and where l is a real parameter, is perhaps the most elemental differential equation in thesciences. The linear ODE (1) represents the dynamics of a process (X(t))t≥0 that converges to a fixed-point level X (∞) = lat an exponential rate r . Indeed, the solution of the linear ODE (1) yields the exponential relaxation process

X(t) = exp (−rt) X(0)+ (1 − exp (−rt)) X (∞) . (2)

5600 I.I. Eliazar, M.H. Cohen / Physica A 391 (2012) 5598–5610

Namely, at time t the value of the process X(t) is a convex combination of its initial value X(0) and its terminal value X (∞),and the convex weight of the initial value is decreasing exponentially at rate r . Exponential relaxations are omnipresent ina multitude of scientific fields including physics, chemistry, biology, engineering, economics, and finance.

In ‘‘real life’’ systems and processes are seldom deterministic. Rather, ‘‘real world’’ systems and processes often involvea noisy term which renders them stochastic. In the context of the linear ODE (1) the incorporation of a white noise termyields the linear stochastic differential equation (SDE)

X(t) = −r (X(t)− l)+ σ W (t), (3)

[28–30]. In the SDE (3) σ is a positive noise-amplitude parameter, and (W (t))t≥0 is the white noise term—the temporalderivative of the Wiener process (W (t))t≥0 (Brownian motion) [31]. The linear SDE (3) – analogous to its deterministiccounterpart, the linear ODE (1) – is perhaps the most elemental SDE in the sciences, with fundamental applications in allthe aforementioned scientific fields (physics, chemistry, biology, engineering, economics, and finance) [32,33].

From a physical-sciences perspective the linear SDE (3) represents random motion in a ‘‘potential well’’ [32,33]. Froman economic–financial perspective the linear SDE (3) represents ‘‘mean reversion’’ [34,35]. In general, the linear SDE (3)describes a motion which is simultaneously and continuously (i) subjected to a deterministic restoring force which drivesthemotion towards the fixed-point level l, and (ii) perturbed by a randomwhite noise which drives themotion off the fixed-point level l. In the physical sciences the stochastic dynamics quantified by the linear SDE (3) are termedOrnstein–Uhlenbeckdynamics [36,37].

The random process (X(t))t≥0 governed by the Ornstein–Uhlenbeck dynamics is asymptotically stationary, and itconverges in law (as t → ∞) to a stochastic steady state. This steady state manifests the ‘‘stochastic balance’’ between thetwo opposing drivers—the deterministic restoring force and the random white noise. The stationary distribution governingthe steady state of the Ornstein–Uhlenbeck dynamics is Gauss with mean l and variance σ 2/2r [32,33]. On a log-linear plotthe probability density function φG(x) (−∞ < x < ∞) of the aforementioned Gauss distribution admits the quadratic from

ln (φG(x)) = cG −rσ 2 (x − l)2 (4)

(cG being a normalizing constant). The Gauss density φG(x) is unimodal, it attains its maximum at the mode xG = l, and it issymmetric around itsmode. The Gauss densityφG(x) is the stationary solution of the Fokker–Planck equation correspondingto the linear SDE (3) [38].

3. Sigmoidal forces and Langevin dynamics

The linear SDE (3) has an implicit underlying assumption: the linearity of the restoring force implies that this force isunbounded. In ‘‘real world’’ systems this implicit assumption is often unrealistic—as the magnitude of the restoring forceis often bounded due to physical limitations. Let the function F(x) (−∞ < x < ∞) represent the restoring force, e.g.,F(x) = r (x − l) in the case of a linear restoring force. The qualitative features of the linear restoring force are smoothness,monotonicity, and crossing of the level zero. If we wish to maintain the qualitative features of the linear restoring force, butyet to impose boundness, we arrive at sigmoidal restoring forces.

A sigmoidal restoring force F(x) = S(x) is a smooth function which is monotone increasing from a lower-bound levellimx→−∞ S(x) = −α to an upper-bound level limx→∞ S(x) = β (α and β being positive parameters). In what follows weset l = S−1(0) to be the fixed-point level at which the sigmoidal force function crosses zero, and set r = S ′(l) to be the slopeof the sigmoidal force function at the fixed-point level l. Smoothness implies that near the fixed-point level l the sigmoidalforce function is approximately linear: S(x) ≈ r (x − l) (as x → l).

Replacing the linear restoring force F(x) = r (x − l) with a general nonlinear restoring force F(x), the linear SDE (3)transforms to the nonlinear SDE

X(t) = −F (X(t))+ σ W (t). (5)

In the physical sciences the stochastic dynamics quantified by the nonlinear SDE (5) are termed Langevin dynamics [17,18]. As in the case of the linear Ornstein–Uhlenbeck dynamics, also the nonlinear Langevin dynamics lead to a stochasticsteady state. Indeed, the random process (X(t))t≥0 governed by the Langevin dynamics is asymptotically stationary, and itconverges in law (as t → ∞) to a stochastic steady state that manifests the ‘‘stochastic balance’’ between the deterministicrestoring force and the random white noise. However, contrary to the linear Ornstein–Uhlenbeck dynamics, the nonlinearLangevin dynamics do not yield a Gauss steady-state distribution.

Let V (x) (−∞ < x < ∞) be a potential function corresponding to the force function F(x), i.e., the potential functionis a primitive of the force function: V ′(x) = F(x). The stationary distribution governing the steady state of the Langevindynamics is quantified by the probability density function

φ(x) = c exp

−2σ 2

V (x)

(6)

(−∞ < x < ∞; c being a normalizing constant). The probability density function φ(x) is the stationary solution of theFokker–Planck equation corresponding to the nonlinear SDE (5) [38].

I.I. Eliazar, M.H. Cohen / Physica A 391 (2012) 5598–5610 5601

In the case of a linear force function F(x) = r (x − l) Eq. (6) yields the Gauss density of Eq. (4): φ (x) = φG(x). In the caseof a sigmoidal force function F(x) = S(x) Eq. (6) yields the Langevin density φ(x) = φS(x) which, on a log-linear plot, isgiven by

ln (φS(x)) = cS −2σ 2

x

lS(u)du (7)

(cS being a normalizing constant).The sigmoidal shape of the force function S(x) implies that the corresponding potential function VS(x) =

xl S(u)du is

U-shaped, attains its global minimum at the fixed-point level l, and satisfies lim|x|→∞ VS(x) = ∞. In turn, the shape ofthe potential function VS(x) implies that the Langevin density φS(x) is unimodal, and that it attains its global maximumat the mode xS = l. The sigmoidal shape of the force function S(x) further implies that its second derivative is bounded:S ′′

= sup−∞<x<∞

S ′′(x) < ∞. Consequently, a second order Taylor expansion of the force function S(x) around the

level l, combined together with Eqs. (4) and (7), implies that

|[ln (φS (x))− cS] − [ln (φG(x))− cG]| ≤

S ′′

3σ 2|x − l|3 (8)

(−∞ < x < ∞). Hence, on a log-linear plot, and up to the normalization constants cS and cG, the Langevin density φS(x) iswell approximated by the Gauss density φG(x) in a neighborhood of their joint mode xS = xG = l.

The approximation of Eq. (8) captures the behavior of the Langevin density φS(x) only near its mode. To characterize thebehavior of the Langevin density φS(x) far off its mode we set α = 2α/σ 2 and β = 2β/σ 2. The boundness of the sigmoidalforce function S(x), combined together with Eq. (7), implies that

ln (φS(x)) ≈

c− − α |x| (as x → −∞),c+ − β |x| (as x → +∞),

(9)

where c− and c+ are associated constants. Hence, the tails of the Langevin density φS(x) are exponential—with exponentα in the limit x → −∞, and with exponent β in the limit x → +∞. Note that if the sigmoidal force function S(x) has abounded magnitude then its lower-bound and upper-bound levels coincide α = β , and consequently the exponents of theLangevin density φS(x) coincide: α = β .

We conclude that shifting from a linear restoring force F(x) = r (x − l) to a sigmoidal restoring force F(x) = S(x)transforms the stochastic dynamics from Ornstein–Uhlenbeck to Langevin—consequently transforming the Gauss densityφG(x) to the Langevin density φS(x). This ‘‘linear-to-sigmoidal force shift’’ has the following affect on the resulting Langevindensity φS(x):• it retains the unimodal shape of the Gauss density, and retains the location of the mode l;• it leaves the center of the Gauss density unchanged—keeping it Gauss in a neighborhood of the mode l;• it thickens the tails of the Gauss density—changing their decay from Gauss to exponential.

4. Geometric Langevin dynamics

The Langevin dynamics – as evident from the nonlinear SDE (5) – are additive dynamics. Indeed, in the nonlinear SDE (5)the dynamical change of themotion (X(t))t≥0 is measured in terms of themotion’s displacements, and is hence additive. Thisunderlying additive dynamical setting is prevalent in the physical sciences. However, there are many dynamical settings– mainly common in economics and in finance [13,14], yet appearing also in the physical sciences [19–21] – in which thedynamical change is measured in terms of the motion’s yields, rather than in terms of the motion’s displacements. Whenthe dynamical change is measured in yields then the underlying dynamical setting ismultiplicative rather than additive.

Transforming the additive dynamical setting of the Langevin dynamics to a multiplicative dynamical setting isstraightforward and is carried out by exponentiation. Indeed, applying the exponential transformation X(t) → Y (t) =

exp (X(t)) to a motion governed by the Langevin dynamics of Eq. (5) (and using Ito’s formula [39]) yields the geometricLangevin dynamics given by the nonlinear SDE

Y (t)Y (t)

= −G (Y (t))+ σ W (t), (10)

where the ‘‘geometric force function’’ is given by G(y) = F (ln(y)) − σ 2/2 (y > 0). Clearly, in the nonlinear SDE (10) thedynamical change of the motion (Y (t))t≥0 is measured in yields—rather than in displacements, as in the nonlinear SDE (5).

Since the motion (X(t))t≥0 is an asymptotically stationary random process which converges in law (as t → ∞) to astochastic steady state—so is the motion (Y (t))t≥0. Let ψ(y) (y > 0) denote the probability density function quantifyingthe stationary distribution governing the steady state of the geometric Langevin dynamics. The connection between thestationary density ψ(y) of the geometric Langevin dynamics and the stationary density φ(x) of the Langevin dynamics isgiven by

ψ(y) = φ (ln(y))1y. (11)

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The probability density function ψ(y) is the stationary solution of the Fokker–Planck equation corresponding to thenonlinear SDE (10) [38].

In the case of a linear force function F(x) = r (x − l) the geometric Langevin dynamics become geometricOrnstein–Uhlenbeck dynamics. The stationary distribution governing the steady state of the geometric Ornstein–Uhlenbeckdynamics is Log-Gauss with parameters l and σ 2/2r (the former being the corresponding Gauss mean, and the latter beingthe corresponding Gauss variance). On a log–log plot the probability density functionψLG(y) = φG (ln(y)) /y (y > 0) of theaforementioned Log-Gauss distribution admits the quadratic from

ln (ψLG(y)) = cLG −rσ 2 (ln(y)− l)2 − (ln (y)− l) (12)

(cLG being a normalizing constant). The Log-Gauss density ψLG(y) is unimodal and skewed, and it attains its maximum atthe mode yLG = exp

l − σ 2/(2r)

.

In the case of a sigmoidal force function F(x) = S(x) the stationary density of the Langevin dynamics is φS(x), and hencethe stationary density of the geometric Langevin dynamics is ψLS(y) = φS (ln(y)) /y. Using Eq. (7) we obtain that, on alog–log plot, the geometric Langevin density ψLS(y) is given by

ln (ψLS(y)) = cLS −

ln(y)

l

1 +

2σ 2

S (u)du (13)

(cLS being a normalizing constant). As in Section 3 we set α = 2α/σ 2 and β = 2β/σ 2. The shape of the geometric Langevindensity ψLS(y) depends on the value of the parameter α as follows: (i) in the parameter range α < 1 the density ψLS(y) ismonotone decreasing and unbounded, and it attains its infinite global maximum at the mode yLS = 0; (ii) at the parametervalueα = 1 the densityψLS(y) ismonotone decreasing and bounded, and it attains its globalmaximumat themode yLS = 0;(iii) in the parameter range α > 1 the densityψLS(y) is unimodal and skewed, and it attains its global maximum at themodeyLS = exp

F−1

−σ 2/2

.

The ‘‘geometric counterpart’’ of Eq. (8) is

|[ln (ψLS (y))− cLS] − [ln (ψLG(y))− cLG]| ≤

S ′′

3σ 2|ln(y)− l|3 . (14)

Hence, on a log–log plot, and up to the normalization constants cLS and cLG, the geometric Langevin density ψLS(y) is wellapproximated by the Log-Gauss densityψLG(y) in a neighborhood of the level y = exp(l). We emphasize that the modes yLGand yLS are always smaller than the level y = exp(l). Note that, in general, the modes yLG and yLS do not coincide; the modesdo coincide if and only if the condition F

l − σ 2/(2r)

= −σ 2/2 is satisfied. For a comprehensive study of the modes of

Langevin and geometric Langevin dynamics the readers are referred to Ref. [40].The ‘‘geometric counterpart’’ of Eq. (9) is

ln (ψLS(y)) ≈

c0 + (α − 1) ln(y) (as y → 0),c∞ − (β + 1) ln(y) (as y → ∞),

(15)

where c0 and c∞ are associated constants. Hence, the tails of the geometric Langevin density ψLS(y) are power-laws—withexponent (α − 1) in the limit y → 0, and with exponent − (β + 1) in the limit y → ∞.

Note that if the sigmoidal force function S(x) has a bounded magnitude then its lower-bound and upper-bound levelscoincide α = β , and consequently the exponents of the geometric Langevin density ψLS(y) are (α − 1) and − (α + 1).In particular, the sum of the exponents is −2. Monotone decreasing probability density functions with such power-lawexponents, and with 0 < α < 1, are encountered in the context of first passage times [22] in anomalous diffusion[23–25]—in which case they are theoretically explained by the continuous time random walk model [26,27]. We furthernote that an analogous structure to that of the double power-law tails of Eq. (15) is widely observed in the context of rankdistributions [41].

We conclude that shifting from a linear restoring force F(x) = r (x − l) to a sigmoidal restoring force F(x) =

S(x) transforms the multiplicative stochastic dynamics from geometric Ornstein–Uhlenbeck to geometric Langevin—consequently transforming the Log-Gauss densityψLG(y) to the geometric Langevin densityψLS(y). This ‘‘linear-to-sigmoidalforce shift’’ has the following affect on the resulting geometric Langevin density ψLS(y):

• it induces a phase transition that takes place as the exponent α crosses unity: in the exponent range α ≤ 1 the shiftchanges the shape of the Log-Gauss density from unimodal to monotone decreasing, and in the exponent range α > 1the shift retains the unimodal shape of the Log-Gauss density—yet it changes the location of the mode;

• it leaves the center of the Log-Gauss density unchanged—keeping it Log-Gauss in a neighborhood of the level exp(l);• it thickens the tails of the Log-Gauss density—changing their decay from Log-Gauss to power-law.

We emphasize that the effect of the ‘‘linear-to-sigmoidal force shift’’ on the stationary density is more elaborate in thecase of the multiplicative geometric Langevin dynamics, than in the case of the additive Langevin dynamics.

I.I. Eliazar, M.H. Cohen / Physica A 391 (2012) 5598–5610 5603

5. An example

To illustrate the results obtained in Sections 3 and 4 consider the following example of a hyperbolic-tangent forcefunction

S(x) = q tanhxp

(16)

(−∞ < x < ∞), where q and p are positive parameters. This force function is sigmoidal, and: (i) its fixed-point level is theorigin l = 0; (ii) its slope at the origin is r = q/p; (iii) its lower-bound and upper-bound levels coincide, and are given byα = β = q. Consequently, we have α = β = 2q/σ 2.

The potential function corresponding to the hyperbolic-tangent force function is

VS(x) = qp lncosh

xp

(17)

(−∞ < x < ∞). The potential function is U-shaped, it attains its global minimum at the origin, it is symmetric around theorigin, and it satisfies lim|x|→∞ VS(x) = ∞.

The Langevin density induced by the hyperbolic-tangent force function is

φS(x) = cS1

cosh

xp

αp (18)

(−∞ < x < ∞).1 The Langevin density φS(x) is unimodal, it attains its global maximum at the mode xS = 0, and it issymmetric around its mode. The tails of the Langevin density φS(x) are exponential: φS(x) ≈ (2αpcS) exp (−α|x|) in thelimit |x| → ∞.

The geometric Langevin density induced by the hyperbolic-tangent force function is

ψLS(y) = cLSyα−1

1 + y2/pαp (19)

(y > 0; the precise value of the normalizing constant is cLS = 2αpcS). In the exponent range α ≤ 1 the density ψLS(y) ismonotone decreasing, and its global maximum is attained at the mode yLS = 0. In the exponent range α > 1 the densityψLS(y) is unimodal and skewed, and it attains its global maximum at the mode

yLS =

α − 1α + 1

p/2

. (20)

The tails of the geometric Langevin density ψLS (y) are power-law: ψLS(y) ≈ cLSyα−1 in the limit y → 0, and ψLS(y) ≈

cLSy−α−1 in the limit y → ∞.The Gauss density φG(x) corresponding to the Langevin density φS(x), and the Log-Gauss density ψLG(y) corresponding

to the geometric Langevin density ψLS(y), are both characterized by the parameters l = 0 and r/σ 2= α/(2p) (the former

being theGaussmean, and the latter being theGauss variance). The globalmaximumof the corresponding Log-Gauss densityψLG(y) is attained at the mode yLG = exp (−p/α). The approximations of Eqs. (8) and (14) hold with the constantS ′′

3σ 2

=2

9√3

α

p2. (21)

6. Stochastic limits

In this section we explore the limiting statistical behavior of the Langevin density φS(x), and of the geometric Langevindensity ψLS(y), in the two following extremal scenarios: (i) the Linear Scenario—in which the slope of the sigmoidal forcefunction at the fixed-point level l is kept constant, whereas the bounds of the force function tend to infinity; (ii) theHeavisideScenario—in which the bounds of the sigmoidal force function are kept constant, whereas the slope of the force function atthe fixed-point level l tends to infinity. In what follows S(x) is the sigmoidal force function introduced in Section 3.

The Linear Scenario is represented by the sigmoidal force function

Sn(x) = nSl +

1n(x − l)

, (22)

1 The precise value of the normalizing constant is cS = (21−αp/p)Γ (αp) /Γ (αp/2)2 .

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where n ≥ 1 is a limit parameter. The characteristics of the sigmoidal force function Sn(x) are: (i) zero-crossing level:S−1n (0) = l; (ii) slope at zero-crossing level: S ′

n(l) = r; (iii) lower-bound and upper-bound levels: limx→−∞ Sn(x) = −nαand limx→∞ Sn(x) = nβ . In the limit n → ∞ the force function Sn(x) converges to the limiting linear force functionS∞(x) = r (x − l)—which represents the restoring force of the Ornstein–Uhlenbeck dynamics. Consequently, in the limitn → ∞ the Langevin density φS(x) converges to the Gauss density φG(x), and the geometric Langevin density ψLS(y)converges to the Log-Gauss density ψLG(y). Noting that

S ′′n

=S ′′

/n, and using Eqs. (8) and (14), it can be shown thatthe aforementioned convergences hold uniformly on bounded domains.

The Heaviside Scenario is represented by the sigmoidal force function Sn(x) = S(nx), where n ≥ 1 is a limit parameter.The characteristics of the sigmoidal force function Sn(x) are: (i) zero-crossing level: S−1

n (0) = l/n; (ii) slope at zero-crossinglevel: S ′

n (l/n) = rn; (iii) lower-bound and upper-bound levels: limx→−∞ Sn(x) = −α and limx→∞ Sn(x) = β . In the limitn → ∞ the force function Sn(x) converges to the Heaviside force function

S∞(x) =

−α (for x < 0),β (for x > 0). (23)

Consequently, in the limit n → ∞ the Langevin density φS(x) converges to the Laplace density

φL(x) =αβ

α + β·

exp (−α |x|) (for x < 0),exp (−β |x|) (for x > 0), (24)

and the geometric Langevin density ψLS(y) converges to the Log-Laplace density

ψLL(y) =αβ

α + β·

yα−1 (for y < 1),y−β−1 (for y > 1).

(25)

Using the sigmoidal shape of the force function S(x) it can be shown that the aforementioned convergences hold uniformlyon domains which are bounded away from the point of discontinuity—the level x = 0 in the Langevin dynamics, and thelevel y = 1 in the geometric Langevin dynamics.

The Laplace density φL(x) is unimodal, and it attains its global maximum at the mode xL = 0. The Laplace density φL(x)is symmetric around its mode if and only if the lower-bound and upper-bound levels coincide α = β—in which case itstail exponents coincide α = β . The shape of the Log-Laplace density ψLL(y) depends on the value of the parameter α asfollows: (i) in the parameter range α < 1 the density ψLL(y) is monotone decreasing and unbounded, and it attains itsinfinite global maximum at the mode yLL = 0; (ii) at the parameter value α = 1 the density ψLL(y) is monotone decreasingand bounded, and it attains its global maximum all along the unit interval; (iii) in the parameter range α > 1 the densityψLL(y) is unimodal and skewed, and it attains its global maximum at the mode yLL = 1.

In the Linear Scenario the limiting force function S∞ (x) is infinitely smooth, whereas in the Heaviside Scenario thelimiting force function S∞(x) is discontinuous at the origin. This discontinuity of the limiting Heaviside force function isinduced to the limiting Laplace density φL(x) – where it is manifested by a cusped maximum, and is further induced to thelimiting Log-Laplace density ψLL(y) – where it is manifested by a cusp at the level y = 1.

Summarizing the results obtained in this section we conclude that:• Sigmoidal force functions span an infinite array of restoring forces ranging between two extremes—the linear force

function characterizing the Linear Scenario, and the Heaviside force function characterizing the Heaviside Scenario.• In the limit of the Linear Scenario the Gauss center of the Langevin density φS(x) takes over the entire real line, and the

Log-Gauss center of the geometric Langevin density ψLS(y) takes over the entire positive half line.• In the limit of the Heaviside Scenario the exponential tails of the Langevin density φS(x) take over the entire real line,

and the power-law tails of the geometric Langevin density ψLS(y) take over the entire positive half line.

Examples of the Linear Scenario and of the Heaviside scenario are depicted, respectively, in Fig. A and in Fig. B. Fig. Aillustrates the Gauss and the Log-Gauss ‘‘central behaviors’’ of, respectively, the Langevin density φS(x) and the geometricLangevin density ψLS(y). Fig. B illustrates the exponential and the power-law ‘‘tail behaviors’’ of, respectively, the Langevindensity φS(x) and the geometric Langevin density ψLS(y).

7. Tails thickening

In Sections 3 and 4 we obtained that shifting from a linear to a sigmoidal force function results in a ‘‘tailsthickening’’ phenomena. Indeed, in the case of Langevin dynamics the ‘‘linear-to-sigmoidal force shift’’ thickens the tails ofthe stationary density φS(x) by changing them from Gauss to exponential, and in the case of geometric Langevin dynamicsthe shift thickens the tails of the stationary density ψLS(y) by changing them from Log-Gauss to power-law.

The thickening of tails is well manifested via moment generating functions in the case of additive stochastic dynamics,and viamoment functions in the case ofmultiplicative stochastic dynamics. Themoment generating function of a probabilitydensity function defined on the real line φ(x) (−∞ < x < ∞) is given by:

φ (θ) =

∞

−∞

exp (θx) φ(x)dx (26)

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Fig. A. The Linear Scenario. In this figure we exemplify the Linear Scenario of Section 6with the hyperbolic-tangent force function S(x) = tanh(x) (and theparameter σ =

√2). (A1) The sigmoidal force function Sn(x), and the limiting linear force function S∞(x) = x. (A2) The corresponding potential function

Vn(x), and the limiting quadratic potential function V∞(x) = x2/2. (A3) The resulting Langevin density Φn(x) = cn exp (−Vn(x)) (cn being a normalizingconstant), and the limiting Gauss density Φ∞(x) = (1/

√2π) exp(−x2/2). (A4) The resulting geometric Langevin density Ψn(y) = Φn (ln(y)) /y, and the

limiting Log-Gauss density Ψ∞(y) = Φ∞ (ln(y)) /y. The figures exemplify the rapid convergence to the limits, and the details of the functions appearingin the figures are given by Eqs. (16)–(19) with parameters q = p = n.

(−∞ < θ < ∞). The moment function of a probability density function defined on the positive half line ψ(y) (y > 0) isgiven by:

ψ(m) =

∞

0ymψ(y)dy (27)

(−∞ < m < ∞).In the case of Langevin dynamics themoment generating function of the stationary density is: (i) convergent on the entire

real line −∞ < θ < ∞ when the restoring force is linear; (ii) convergent on the interval −α < θ < β – and divergentoutside of this interval – when the restoring force is sigmoidal or Heaviside. In the case of geometric Langevin dynamics themoment function of the stationary density is: (i) convergent on the entire real line −∞ < m < ∞ when the restoring forceis linear; (ii) convergent on the interval −α < m < β – and divergent outside of this interval – when the restoring force issigmoidal or Heaviside.

The thickening of tails implies an ‘‘ordering of randomness’’ regarding the stationary distributions we encountered.Indeed, considering a stationary distribution quantified by a given probability density function: the ‘‘lighter’’ the tailsof the density – the more tame the distribution’s fluctuations; the ‘‘heavier’’ the tails of the density – the more feralthe distribution’s fluctuations. Mandelbrot categorized probability distributions into seven classes of randomness rangingfrom ‘‘mild randomness’’ to ‘‘wild randomness’’ [42]. Recently, a Mandelbrot-style classification of randomness – based onmoment generating functions and on moment functions – was introduced and applied to Langevin and geometric Langevindynamics [43]. According to this classification of randomness the ‘‘ordering of randomness’’ of the stationary distributionswe encountered is the following:

φG(x) ≺ {φS (x) , φL(x)} ≺ ψLG(y) ≺ {ψLS(y), ψLL(y)} . (28)

More specifically [43]: (i) The Gauss densityφG(x) belongs to the class of ‘‘ultra-mild randomness’’ which is characterizedbymoment generating functions converging for all θ . (ii) The Langevin density φS(x) and the Laplace density φL(x) belong tothe class of ‘‘mild randomness’’ which is characterized bymoment generating functions converging in a finite neighborhoodof the origin. (iii) The Log-Gauss density ψLG(y) belongs to the class of ‘‘borderline randomness’’ which is characterized bymoment generating functions diverging for all θ > 0, and bymoment functions converging for allm > 0. (iv) The geometricLangevin densityψLS(y) and the Log-Laplace densityψLL(y) belong to the class of ‘‘wild randomness’’ which is characterized

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yy

10.80.60.40.20–0.2–0.4–0.6–0.8–1x

x

0.81

0.60.40.2

0–0.2–0.4–0.6–0.8

–1

1

0

2

3

4

6

5

y

0 2 4 6–2–4–6

y

0

0.2

0.3

0.4

0.5

0.1

–5 –4 –3 –2 –1 0 1 2 3 4 5

1 2 3 3.50 0.5 1.5 2.5

x

x

0

0.2

0.3

0.4

0.6

0.5

0.1

Fig. B. The Heaviside Scenario. In this figure we exemplify the Heaviside Scenario of Section 6 with the hyperbolic-tangent force function S(x) = tanh(x)(and the parameter σ =

√2). (B1) The sigmoidal force function Sn(x), and the limiting Heaviside force function S∞(x) = sign(x). (B2) The corresponding

potential function Vn(x), and the limiting absolute-value potential function V∞(x) = |x|. (B3) The resulting Langevin density Φn(x) = cn exp (−Vn(x))(cn being a normalizing constant), and the limiting Laplace density Φ∞(x) = (1/2) exp(− |x|). (B4) The resulting geometric Langevin density Ψn(y) =

Φn (ln(y)) /y, and the limiting Log-Laplace density Ψ∞(y) = Φ∞ (ln(y)) /y. The figures exemplify the rapid convergence to the limits, and the details ofthe functions appearing in the figures are given by Eqs. (16)–(19) with parameters q = 1 and p = 1/n.

Table 1In this table we summarize the randomness classifications of the probability distributions emanating fromLangevin and geometric Langevin dynamics. The left column of the table specifies the force function applied.The middle column of the table specifies the stationary density obtained – in the context of Langevindynamics – and its randomness classification. The right column of the table specifies the stationary densityobtained – in the context of geometric Langevin dynamics – and its randomness classification.

Force Langevin dynamics Geometric Langevin dynamics

Linear Gauss density φG(x) Log-Gauss density ψLG(y)Ultra-mild randomness Borderline randomness

Heaviside Laplace density φL(x) Log-Laplace density ψLL(y)Mild randomness Wild randomness

Sigmiodal Langevin density φS(x) Geometric Langevin density ψLS(y)Mild randomness Wild randomness

by moment functions converging in a finite positive neighborhood of the origin. Table 1 summarizes the key details of thesix aforementioned stationary distributions.

8. Financial markets

The common and widely applied statistical models for tradable financial assets are based on the assumptionthat the distribution of prices is Log-Gauss [13]. The quintessential example of the Log-Gauss assumptions is theMerton–Black–Scholes option pricing formula [15,16]. However, the true statistical behavior of tradable financial assetsis not Log-Gauss, and the distribution of prices possesses power-law tails [14].

The discrepancy between the Log-Gauss behavior of price-models and the power-law behavior of true prices is ofthe utmost importance. The Log-Gauss distribution – as noted in Section 7 – has converging moments of all orders, anddisplays ‘‘borderline randomness’’. On the other hand, probability distributions with power-law tails fail to have convergingmoments of all orders, and display ‘‘wild randomness’’. This distinction is far from being a mere theoretical detail. Rather,this distinction has dramatic practical implications. Indeed, according to the Log-Gauss assumption the crashes of theS&P financial index during the twentieth century were statistically impossible—yet these crashes did very well occur in

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reality [14]. The dramatic difference between the tame fluctuations of the Log-Gauss distribution and the feral fluctuationsof probability distributions with power-law tails was adamantly proclaimed and advocated by Mandelbrot—pointing outand emphasizing the diametric distinction between ‘‘mild’’ and ‘‘wild’’ types of randomness [42,44].

The geometric Langevin densityψLS(y) derived in Section 4 provides a conceptual ‘‘Langevin toy model’’ to the behaviorof financial markets:

• In ‘‘normal times’’ – characterized by y ≈ exp(l) – markets display a ‘‘near-equilibrium behavior’’: the underlyingrestoring forces are linear, these linear forces keep markets well at balance, and the resulting price fluctuations are tameand approximately Log-Gauss.

• In ‘‘abnormal times’’ – characterized by y → 0 and y → ∞ – markets display an ‘‘off-equilibrium behavior’’: theunderlying restoring forces are constant, these constant forces cannot quickly reinstate the markets back to balance,and the resulting price fluctuations are feral and governed by power-laws.

We emphasize that the aforementioned ‘‘abnormal times’’ occur both during financial debacles (y → 0), and duringfinancial bobbles (y → ∞). In these ‘‘abnormal times’’ individual rationality is replaced by irrational herd behavior [45,46],and the underlying restoring forces are too week to steadfastly drive the markets back to rationality and balance. Thesociological phenomena of herd behavior – in this conceptual Langevin toy model – is manifested by the boundness of theunderlying restoring forces. In game theory and behavioral economics researchers often introduce the notion of ‘‘boundedrationality’’ to explain irrational behavior [47,48]. With the conceptual Langevin toy model in mind we can thus draw ananalogy between the notion of force in the physical sciences and the notion of rationality in the socioeconomic sciences.

9. The DPLN model

As noted above, Geometric Brownian Motion (GBM) is the quintessential model of stochastic multiplicative growthprocesses, is commonly applied in economics and finance [13], and underlies the Merton–Black–Scholes option pricingformula [15,16]. The stochastic dynamics of GBM are governed by the SDE

Z(t)Z(t)

= µ+ σ W (t) . (29)

In Eq. (29) (Z(t))t≥0 is the GBM under consideration, µ is the GBM’s real-valued ‘‘drift parameter’’, σ is the GBM’s positive-valued ‘‘volatility parameter’’, and (W (t))t≥0 is a driving white noise—the temporal derivative of the Wiener process(W (t))t≥0 [31]. The GBM dynamics governed by the SDE (29) are a ‘‘non-equilibrium counterpart’’ of the geometricLangevin dynamics governed by the SDE (10). Solving the SDE (29) (using Ito’s formula [39]) yields the following explicitrepresentation of GBM:

Z(t) = Z(0) expµ−

12σ 2

t + σW (t)

. (30)

With no loss of generality we henceforth consider the initial condition Z(0) = 1.GBM is a growth process which does not converge in law (as t → ∞) to a stochastic steady state. Thus, we stop the GBM

after at a given time T , and focus on the random level Z(T ) attained. If T = Tdet is a deterministic time then the random levelZ(Tdet) is Log-Gauss distributed—its logarithm ln(Z(Tdet)) being Gauss distributed with mean (µ − σ 2/2)Tdet and varianceσ 2Tdet. Consider now the stopping time T = Texp to be an exponentially distributed random variable with mean 1/λ, whichis independent of the Wiener process (W (t))t≥0. The parameter λ is the ‘‘killing rate’’ of the exponential time Texp. In thiscase the random level Z(Texp) is Log-Laplace distributed—its logarithm ln(Z(Texp)) being Laplace distributed. Indeed, set

ρ± =

µ

σ 2−

12

+

2λσ 2

±

µ

σ 2−

12

. (31)

Then [11]: the probability density functions of the random variables ln(Z(Texp)) and Z(Texp) are given, respectively, by theLaplace and Log-Laplace densities of Eqs. (24) and (25), with exponents α = ρ+ and β = ρ−.

In the DPLN model GBM is run for a deterministic time period, and thereafter run for an additional exponential timeperiod [3,11] (in these different time periods the GBM might even have different drift and volatility parameters). Namely,in the DPLN model we have T = Tdet + Texp, where Tdet and Texp are as above. Consequently, the random level Z(T ) is equal,in law, to the product of two independent random factors: (i) a first factor which is equal in law to the aforementionedrandom level Z(Tdet); (ii) a second factor which is equal in law to the aforementioned random level Z(Texp). The randomlevel Z(T ) displays a Log-Gauss–Pareto composite structure. Indeed, the probability density function of the random levelZ(T ) is Log-Gauss at its central body, follows an asymptotic power-law with exponent (α − 1) near the origin, and followsan asymptotic power-law with exponent − (β + 1) near infinity [3,11].

Thus, the geometric Langevin dynamics and the GBM dynamics facilitate diametrically different approaches yieldingthe Log-Gauss–Pareto composite structure. The geometric Langevin dynamics establishes an ‘‘equilibrium model’’: therandom motion emanating from these dynamics converges in law to a stochastic steady state—which is governed by the

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Table 2In this tablewe summarize the two diametric equilibrium and non-equilibrium approaches yielding the Log-Gauss–Pareto composite statistical structure. The left column of the table specifies the ‘‘target statistics’’ weaim at attaining. The equilibrium approach is based on the steady state distribution of geometric Langevindynamics, and is given in the middle column—specifying the type of force function that needs to be appliedin order to yield a desired ‘‘target statistics’’. The non-equilibrium approach is based on the stopping ofgeometric Brownian motion (GBM), and is given in the right column—specifying the type of the stoppingtime that needs to be applied in order to yield a desired ‘‘target statistics’’.

Target statistics Equilibrium approach:Geometric Langevin dynamics

Non-equilibrium approach:GBM dynamics

Log-Gauss Linear force function Deterministic stopping time

Log-Laplace Heaviside force function Exponential stopping time

Log-Gauss–Paretocomposite structure

Sigmoidal force function Sum of deterministic and exponentialstopping times

Log-Gauss–Pareto composite structure when sigmoidal force functions are applied. The GBM dynamics establishes a ‘‘non-equilibrium model’’: the random motion emanating from these dynamics is a random growth process whose random level– at a stopping time T – is governed by the Log-Gauss–Pareto composite structure when the stopping time is a sum ofa deterministic and an exponential random variables. A comparative summary of these different approaches is given inTable 2.

10. Universality

All the stochastic dynamics discussed so forth were driven by a white noise term (W (t))t≥0—the temporal derivative ofthe Wiener process (W (t))t≥0 [31]. The Wiener process is a stochastic process whose increments – which are independentand stationary – are Gauss distributed. The underlying Gauss statistics of the Wiener process induce the Gauss and Log-Gauss statistics encountered in following settings we considered: Ornstein–Uhlenbeck dynamics, Langevin dynamics withsigmoidal forces, geometric Ornstein–Uhlenbeck dynamics, geometric Langevin dynamics with sigmoidal forces, GBMdynamics, and the DPLN model.

This ‘‘Gauss induction’’ raises serious doubts regarding the broad applicability of the aforementioned settings. Indeed,having used a very specific type of noise – white noise – might very well render the results obtained to be rather particularand not widely applicable. In other words, the white noise assumption – implicitly underlying all settings considered –might be a restrictive and a generally inapplicable assumption. Gladly so, these serious concerns are waived by Donsker’sFunctional Central Limit Theorem (FCLT) [49,50].

Donsker’s FCLT asserts that the Wiener process (W (t))t≥0 is the universal stochastic scaling limit of random walks withfinite-variance jumps. Consider a random walk process (ξ(t))t≥0 whose jumps are governed by a probability distributionwith zero mean and unit variance. Given a limit parameter n ≥ 1, ‘‘renormalize’’ the random walk as follows: speed upthe time of the random walk by the factor n, and scale the jumps of the random walk by the factor

√n. The renormalized

random walk process (ξn(t))t≥0 is thus given by ξn(t) = ξ(nt)/√n. Donsker’s FCLT asserts that the renormalized random

walk process (ξn(t))t≥0 converges in law, as n → ∞, to the Wiener process (W (t))t≥0. The key feature of Donsker’s FCLT isits universality. Indeed, the Wiener-process limit turns out to be invariant with respect to the particular distribution of therandom walk’s jumps.

From a physical perspective the FCLT’s universality is manifested by transcendence from the micro-scale to the macro-scale: on amicroscopic scale random-walk statistics can have infinitelymany degrees of freedom, but on amacroscopic scalethere is only one universal type of random-walk statistics—theWiener-process statistics. From a mathematical perspectivethe FCLT’s universality manifests invariance with respect to the probability distribution governing the jumps of the randomwalk. This probability distribution, in turn, is quantified by its cumulative distribution function ϕ(x) (−∞ < x < ∞)—a function which is monotone increasing from the lower-bound level limx→−∞ ϕ(x) = 0 to the upper-bound levellimx→∞ ϕ(x) = 1.

It is illuminating to compare Langevin dynamicswith sigmoidal force functions andDonsker’s FCLT. From amathematicalperspective the ‘‘input’’ in the case of the Langevin dynamics considered is the sigmoidal force function S(x), and the‘‘input’’ in the case of Donsker’s FCLT is the cumulative distribution function ϕ(x). Both functions S(x) and ϕ(x) are definedon the real line (−∞ < x < ∞), and are sigmoidal in shape. The force function S(x) is monotone increasing from thelevel −α to the level β , and is smooth. The distribution function ϕ(x) is monotone increasing from the level 0 to the level1, and satisfies the moment conditions

∞

−∞xϕ(dx) = 0 and

∞

−∞x2ϕ (dx) = 1. The ‘‘output’’ of Langevin dynamics is the

Langevin density φS(x), and the ‘‘output’’ of Donsker’s FCLT is the limitingWiener process (W (t))t≥0. In the case of Langevindynamics the ‘‘output’’ is invariant with respect to the ‘‘input’’: the Langevin density φS(x) is Gauss around its mode andis exponential in its tails—regardless of the details of the sigmoidal force function S(x). In the case of Donsker’s FCLT the‘‘output’’ is invariant with respect to the ‘‘input’’: the limiting random walk is the Wiener process (W (t))t≥0—regardless ofthe details of the cumulative distribution function ϕ(x). Thus, from a mathematical perspective, Langevin dynamics withsigmoidal force functions and Donsker’s FCLT are two ‘‘universal mechanisms’’ that share a common intrinsic universality-generating functional structure.

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We conclude that in the context of Langevin and geometric Langevin dynamics with sigmoidal force functions there aretwo universal mechanisms operating jointly as follows:

• Donsker’s FCLT generates, from general micro-scale noises, a universal macro-scale white noise—which drives theLangevin dynamics.

• General sigmoidal force functions, applied on the Langevin dynamics, generate a stationary Langevin density with auniversal structure—Gauss around its mode, and exponential in its tails.

• Shifting, via exponentiation, from additive to multiplicative stochastic dynamics yields a stationary geometric Langevindensity with a universal structure—Log-Gauss around its center, and power-law in its tails.

Last, we note that Donsker’s FCLT does not hold in the case of random walks with infinite-variance jumps. Indeed, theuniversal stochastic scaling limits of randomwalks with infinite-variance jumps are Lévy processes (rather than theWienerprocess) [51,52]. Langevin dynamics driven by Lévy noises – the temporal derivative of Lévy processes – display profoundlydifferent statistical behavior than Langevin dynamics driven by white noise, and attracted significant interest in the contextof anomalous diffusion [53–62]. The study of Langevin dynamicswith sigmoidal force functions andwith Lévy driving noisesis beyond the scope of this paper and is a matter of further research.

11. Conclusion

In this paper we established a Langevin explanation to the emergence of the Log-Gauss–Pareto composite statisticalstructure. This composite structure is characterized by probability density functions which are Log-Gauss in their centralbody, and follow power-law decay in their tails. The Log-Gauss–Pareto composite statistical structure is observed in variouscomplex systems, and its quintessential example is the distribution of wealth in human populations. On a logarithmic scalethis composite structure displays a Gauss-Exponential statistical structure—characterized by probability density functionswhich are Gauss in their central body, and follow exponential decay in their tails.

Our starting point was the elemental Ornstein–Uhlenbeck dynamics, which are omnipresent in the sciences. TheOrnstein–Uhlenbeck dynamics are characterized by linear restoring forces, and yield Gauss stationary distributions. Whenphysical limitations impose a boundness constrain on the restoring forces, then the force functions shift from linearto sigmoidal. Consequently, the linear Ornstein–Uhlenbeck dynamics shift to nonlinear Langevin dynamics. We showedthat the ‘‘linear-to-sigmoidal’’ shift of force functions transforms the corresponding stationary distributions – of theOrnstein–Uhlenbeck and Langevin dynamics – from Gauss to the aforementioned Gauss-Exponential composite statisticalstructure.

Moving on from additive stochastic dynamics to multiplicative stochastic dynamics further led us to the Log-Gauss–Pareto composite statistical structure. In additive stochastic dynamics the dynamical change is measured in termsof the displacements of the motion considered, whereas in multiplicative stochastic dynamics the dynamical changeis measured in terms of the motion’s yields. Exponentiation transforms the additive Ornstein–Uhlenbeck dynamicsto the multiplicative geometric Ornstein–Uhlenbeck dynamics, and transforms the additive Langevin dynamics to themultiplicative geometric Langevin dynamics. We showed that the ‘‘linear-to-sigmoidal’’ shift of force functions transformsthe corresponding stationary distributions – of the geometric Ornstein–Uhlenbeck and geometric Langevin dynamics – fromLog-Gauss to the aforementioned Log-Gauss–Pareto composite statistical structure.

The Gauss-Exponential and Log-Gauss–Pareto composite statistical structures emerge invariantly with respect to thespecific details of the underlying sigmoidal force functions.We further explored various aspects of these composite statisticalstructures: (i) their stochastic limits yielding the Gauss and Log-Gauss stationary distributions on one extreme, and yieldingthe Laplace and Log-Laplace stationary distributions on the other extreme; (ii) the ‘‘tails thickening’’ phenomenon inducedby the ‘‘linear-to-sigmoidal’’ shift of force functions, and its statistical implications; (iii) the application of geometricLangevin dynamics, with sigmoidal force functions, as a conceptual toy model for the behavior of financial markets. Inaddition, we compared the Langevin model and the DPLN model – two diametrically different approaches generating theGauss-Exponential and Log-Gauss–Pareto composite statistical structure – the former being an ‘‘equilibrium model’’ andthe latter being a ‘‘non-equilibrium model’’.

Last, we addressed the universality of the Langevin model. We concluded that the Langevin model combines thesimultaneous operation of two ‘‘universality mechanisms’’: (i) a mechanism that ‘‘feeds’’ the Langevin and geometricLangevin dynamics with a macro-scale white noise—invariantly with respect to the details of the micro-scale noise; (ii) amechanism that produces the Gauss-Exponential and Log-Gauss–Pareto composite statistical structures—invariantly withrespect to the details of the sigmoidal force functions applied. Moreover, we showed that, from amathematical perspective,these two universality mechanisms are analogous.

The DPLN model, together with the Langevin model presented herein, provide a theoretical umbrella explaining theemergence of the Gauss-Exponential and Log-Gauss–Pareto composite statistical structures. The DPLN model provides a‘‘non-equilibrium explanation’’, and applies in the context of Brownian motion and geometric Brownian motion. The modelpresented herein provides an ‘‘equilibrium explanation’’, and applies in the context of Langevin and geometric Langevindynamics.We hope that this paperwill well-serve scientists to properly and adequately explain Gauss-Exponential and Log-Gauss–Pareto composite statistical structures—once they are statistically encountered in the outputs of complex systems.

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References

[1] V. Pareto, Cours d’économie Politique, Droz, Geneva, 1896.[2] A. Chatterjee, S. Yarlagadda, B.K. Chakrabarti (Eds.), Econophysics of Wealth Distributions, Springer, Milan, 2005.[3] W.J. Reed, M. Jorgensen, Comm. Statist. Theory Methods 33 (2004) 1733.[4] K. Cooray, M.M.A. Ananda, Scand. Actuar. J. 5 (2005) 321.[5] D.P.M. Scollnik, Scand. Actuar. J. 1 (2007) 20.[6] M. Pigeon, M. Denuit, Scand. Actuar. J. 3 (2011) 177.[7] M. Mitzenmacher, Internet Math. 1 (2004) 226.[8] M.E.J. Newman, Contemp. Phys. 46 (2005) 323.[9] A. Clauset, C.R. Shalizi, M.E.J. Newman, SIAM Rev. 51 (2009) 661.

[10] M. Mitzenmacher, Internet Math. 1 (2004) 305.[11] W.J. Reed, Physica A 319 (2003) 469.[12] W.J. Reed, B.D. Hughes, Phys. Rev. E 66 (2002) 067103.[13] J.C. Hull, Options, Futures, and Other Derivatives, sixth ed., Prentice-Hall, London, 2005.[14] R.N. Mantegna, H.E. Stanley, Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2007.[15] R.C. Merton, Bell J. Econ. Manage. Sci. 4 (1973) 141.[16] F. Black, M. Scholes, J. Polit. Econ. 81 (1973) 637.[17] P. Langevin, C. R. Math. Acad. Sci. Paris 146 (1908) 530.[18] W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, second ed., World Scientific, Singapore, 2004.[19] K. Lindenberg, K.E. Shuler, V. Seshadri, B.J. West, in: T. Bharucha-Reid (Ed.), Probabilistic Analysis and Related Topics, vol. 3, Academic Press, New York,

1983.[20] J.D. Ramshaw, K. Lindenberg, J. Stat. Phys. 45 (1986) 295.[21] I. Eliazar, J. Klafter, Physica A 367 (2006) 106.[22] S. Redner, A Guide to First-Passage Processes, Cambridge University Press, Cambridge, 2001.[23] R. Metzler, J. Klafter, Phys. Rep. 339 (2000) 1.[24] J. Klafter, I.M. Sokolov, Phys. World 18 (2005) 29.[25] I.M. Sokolov, J. Klafter, Chaos 15 (2005) 026103.[26] H. Scher, E.W. Montroll, Phys. Rev. B 12 (1975) 2455.[27] H. Scher, G. Margolin, R. Metzler, J. Klafter, B. Berkowitz, Geophys. Res. Lett. 29 (2002) 1061.[28] A. Friedman, Stochastic Differential Equations and Applications, Dover, New York, 2006.[29] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, sixth ed., Springer, New York, 2010.[30] L. Arnold, Stochastic Differential Equations: Theory and Applications, Dover, New York, 2011.[31] N. Wiener, J. Math. Phys. 2 (1923) 131.[32] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, third ed., North-Holland, Amsterdam, 2007.[33] Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer, New York, 2009.[34] J.M. Poterba, L.H. Summers, J. Fin. Econ. 22 (1988) 27.[35] R. Balvers, Y. Wu, Erik Gilliland, J. Finance 55 (2000) 745.[36] G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36 (1930) 823.[37] J.L. Doob, Ann. of Math. 43 (1942) 351.[38] H. Risken, The Fokker–Planck Equation: Methods of Solutions and Applications, Springer, New York, 1996.[39] K. Ito, Mem. Amer. Math. Soc. 4 (1951) 1.[40] I. Eliazar, M.H. Cohen, The misconception of mean reversion, 2012 (submitted for publication).[41] G. Martinez-Mekler, PLoS One 4 (2009) e4791.[42] B.B. Mandelbrot, Fractals and Scaling in Finance, Springer, New York, 1997.[43] I. Eliazar, M.H. Cohen, From shape to randomness: a classification of Langevin stochasticity, 2012 (submitted for publication).[44] B. Mandelbrot, N.N. Taleb, in: F. Diebold, N. Doherty, R. Herring (Eds.), The Known, the Unknown and the Unknowable in Financial Institutions,

Princeton University Press, Princeton, 2010, pp. 47–58.[45] D.S. Scharfstein, J.C. Stein, Amer. Econ. Rev. 80 (1990) 465.[46] R. Cont, J.P. Bouchaud, Macroecon. Dyn. 4 (2000) 170.[47] H. Simon, in: C.B. McGuire, R. Radner (Eds.), Decision and Organization, North-Holland, Amsterdam, 1972 (Chapter 8).[48] A. Rubinstein, Modeling Bounded Rationality, MIT Press, Boston, 1998.[49] M.D. Donsker, Mem. Amer. Math. Soc. 6 (1951) 1.[50] W. Whitt, Stochastic-Process Limits, Springer, New York, 2002.[51] A. Janicki, A. Weron, Simulation and Chaotic Behavior of Stable Stochastic Processes, Marcel Dekker, New York, 1994.[52] I. Eliazar, M.F. Shlesinger, J. Phys. A 45 (2012) 162002.[53] S. Jespersen, R. Metzler, H.C. Fogedby, Phys. Rev. E 59 (1999) 2736.[54] A.V. Chechkin, et al., Chem. Phys. 284 (2002) 233.[55] A.V. Chechkin, et al., Phys. Rev. E 67 (2003) 010102(R).[56] A.V. Chechkin, et al., Phys. Rev. E 72 (2005) 010101(R).[57] I. Eliazar, J. Klafter, J. Stat. Phys. 111 (2003) 739.[58] I. Eliazar, J. Klafter, J. Stat. Phys. 119 (2005) 165.[59] I. Eliazar, J. Klafter, J. Phys. A 40 (2007) F307.[60] B. Dybiec, E. Gudowska-Nowak, I.M. Sokolov, Phys. Rev. E 76 (2007) 041122.[61] M. Magdziarz, A. Weron, Phys. Rev. E 75 (2007) 056702.[62] M. Magdziarz, Physica A 387 (2008) 123.