Long Term Risk: An Operator Approach∗

Lars Peter Hansen Jose Scheinkman

November 22, 2005

Abstract

We build a family of valuation operators indexed by the increment of time betweenthe payoff date and the current period value. These operators are necessarily related bywhat is known as the semigroup property or the Law of Iterated Values. The operatorformulation we develop provides a way to link short term risk adjustments to whathappens in the medium and long term. We apply this apparatus to give a precisenotion of a long term risk-return tradeoff.

1 Introduction

This paper develops an operator formulation of asset pricing. As in previous research, wemodel asset valuation using operators that assign prices today to payoffs in future dates.Since these operators are defined for each payoff date, we build an indexed family of suchpricing operators. This family is referred to as a semigroup because of the manner in whichthe operators are related to one another. It is the evolution of these operators as the payoffdate changes that interests us. In this paper we study this evolution using a continuous-timeframework, although important aspects of our analysis are directly applicable to discretetime specifications. Our analysis is made tractable by assuming the existence of a Markovstate that summarizes the information in the economy pertinent for valuation. The operatorswe use apply to functions of this Markov state.

A semigroup is typically modeled in terms of a generator, an operator that captureslocal behavior or the evolution over a short period of time. The entire semigroup can beconstructed from this generator via an exponential formula. While continuous-time modelsachieve simplicity by characterizing behavior over small time increments, operator methodshave promise for enhancing our understanding of the connection between short run and longrun behavior.

In financial economics risk return tradeoffs show how expected rates of return over smallintervals are altered as we change the exposure to the underlying shocks that impinge on the

∗This draft is preliminary and incomplete. Comments from Vasco Carvalho were very helpful in preparingthis draft.

1

economy. In continuous time modeling, the length of interval is driven to zero to deduce alimiting relationship. In the operator version of such an analysis, this limiting relationshipis reflected in the generators used to depict values. We demonstrate the potency of operatormethods by interpreting the dominant eigenvalues of valuation operators computed for alter-native stochastic growth processes. As we alter the underlying stochastic growth componentsof cash flows, we change their long run risk exposure. This is reflected by how we alter theconstruction of the valuation operators to accommodate stochastic growth. This alterationturns out to be informative. Dominant eigenvalues encode well defined adjustments for riskin the long run. By changing growth processes we trace out a long run counterpart to arisk-return tradeoff familiar from the local analysis.

The remainder of the paper is organized as follows. In sections 2 and 3 we develop someof the mathematical preliminaries pertinent for our analysis. Specifically, in section 2 weintroduce the reader to the notion of a semigroup and a generator of that semigroup, and insection 3 we introduce the reader to the concepts of an additive and a multiplicative func-tional constructed as functions of a Markov process. Both functionals are crucial ingredientsto what follows. In section 4 we construct a pricing semigroup used to evaluate contingentclaims written on the Markov state indexed by the elapsed time between trading date of thepayoff date. In sections 5 and 6 we define two specific multiplicative functionals that arecentral to our analysis: valuational functionals and reference growth functionals. Valuationfunctionals are used to construct returns over intervals of any horizon. Reference growthfunctionals give benchmarks for building long term risk return tradeoffs. They feature themartingale components of growth and abstract from the transient components. In section8 we introduce principal eigenvalues and functions and demonstrate their role long run val-uation and growth. Finally, in sections 9 and 10 we discuss sufficient conditions for theexistence of principal eigenvalues needed to support our analysis.

2 Semigroups

Let L be a Banach space with norm ‖ · ‖, and let {Tt : t ≥ 0} be a family of operators on L.The operators in these family are linked according to the following property:

Definition 2.1. A family of linear operators {Tt : t ≥ 0} is a one-parameter semigroup ifT0 = I and Tt+s = TtTs for all s, t ≥ 0.

From a forecasting perspective, these operators are conditional expectations operators andthis link typically follows from the Law of Iterated Expectations restricted to Markovprocesses. We will also use such families of operators to study valuation and pricing. Froma pricing perspective, the semigroup property follows the Law of Iterated Values that holdswhen there is frictionless trading at intermediate dates.

Definition 2.2. The semigroup {Tt : t ≥ 0} is strongly continuous if for any ψ ∈ L:

limt↓0‖Ttψ − ψ‖ = 0

2

for some Banach space L.

Strong continuity is known to imply an exponential bound on the growth of the semi-group:

‖Tt‖ ≤ K exp(δt).

for some K ≥ 1 and some positive δ. Later we will give a more refined characterization.Strong continuity of the semigroup is also known to imply the existence of the generator Uof the semigroup. This generator is a closed operator defined on a dense subset, D(U), of Las:

Uψ = limt↓0

Ttψ − ψ

t.

[See Ethier and Kurtz (1986) Corollary 1.6 on page 10.] The operator U is referred to as agenerator because the pricing semigroup may be constructed from U . This construction usesthe exponential formula:

Tt = exp(tU)

which is defined rigorously through the Yosida approximation.1

We will often impose further restrictions on semigroups such as:

Definition 2.3. The semigroup {Tt : t ≥ 0} is positive if for any t ≥ 0, Ttψ ≥ 0 wheneverψ ≥ 0.

3 Markov and Related Processes

Let {Xt : t ≥ 0} be a continuous time Markov process on a state space D0 with a stationarytransition law. We will sometimes assume that this process is stationary and ergodic. LetFt be completion of the sigma algebra generated by {Xu : 0 ≤ u ≤ t}.

3.1 Feller Semigroup

There are alternative ways to construct a Markov process. We sketch the operator approachon a specific function space to build a Feller process, but there are other possible constructionswith similar consequences. The resulting representation we will display of the generator ofa mixed jump diffusion process will be used in our subsequent analysis.

Let D0 be locally compact and separable. We add a point to this space, which we callinfinity, and denote by D the augmented space. When D0 is compact, the point infinity isan isolated point used as a termination state. When D0 is not compact, we use a one pointcompactification in which infinity is the additional point. Set D = D0

⋃∞ and take a set inD to be open if either it is an open set in D0 or the complement of a compact set in D0. LetC denote the space of continuous functions mapping D into R equipped with the sup-norm:‖ · ‖ and let C0 denote the space of continuous functions φ : D0 → R, that can be extended

1See Ethier and Kurtz (1986) page 12 for a formal construction.

3

to a function in C, by setting φ(∞) = 0. We will use the same notation, say φ, to depict afunction in C or its restriction to the domain D0.

Example 3.1. Let D0 denote an open subset of Rn. Then a continuous function φ is in C0

provided that (a) if xj → x, xj ∈ Rn, x 6∈ D0, then φ(xj) → 0, and (b) if ||xj|| → ∞, thenφ(xj) → 0.

Example 3.2. Let D0 be compact. Then C0 is the space of continuous functions on D0.Any such function φ may be extended to the isolated point ∞ in a continuous fashion bydefining this function to be zero at ∞.

Next we introduce a strongly continuous, positive semigroup {Tt : t ≥ 0} on C0. Theseoperators will serve as conditional expectation operators over a time interval t for the con-structed Markov process. Accordingly, they are restricted to be (weak) contractions for eacht. Typically, we think of conditional expectation operators as mapping unit functions intounit functions. A function that is one in all states except ∞ is often not in C0, however.Since the semigroup is positive, we consider an approximation to such a function from belowin the following definition.

Definition 3.1. A semigroup {Tt : t ≥ 0} is conservative if

supφ∈C0,0≤φ≤1

Ttφ = 1.

A Feller semigroup is a strongly continuous, positive, contraction semigroup on C0 that isconservative. For such a semigroup there corresponds a Markov process for such a semigroup,and there are known ways to represent generator of a Feller semigroup. Suppose that D0 is anonempty open set contained in Rn and suppose that the domain of the generator containsC∞

K , the space of infinitely differentiable functions with a compact domain. We use thefollowing object to represent the generator of Feller semigroup:

a) η(dy|x)., a positive Radon measure defined on Rn − {x}, that is a Borel measure that isfinite in compact sets of Rn − {x}.

b) a nonnegative function α;

c) an n-dimensional vector µ of functions;

d) a (pointwise) positive, semidefinite matrix Σ of functions .

Given a a strongly continuous contraction semigroup, and any relatively compact open setO containing x there exists a (α, µ, Σ, η) such that for each function f ∈ C2

K , the generator:

Uφ(x) = − α(x)φ(x) + µ(x) · ∂φ(x)

∂x+

1

2trace

(Σ(x)

∂2φ(x)

∂x∂x′

)

+

∫

Rn−{x}

[φ(y)− φ(x)− 1O(x)(y − x) · ∂φ(x)

∂x

]η(dy|x), (1)

4

where 1O denotes the indicator function of a set O[see Theorem 1.13 of Revuz and Yor(1994)]. Thus we may represent the generator in terms of a four objects (α, µ, Σ, η) The termα corresponds to the possibility of the process being “killed” and is often assumed to be zero.The measure η is used to introduce jumps or sample path discontinuities into the analysis.When this measure is zero, the process is of the diffusion type with drift µ and diffusionmatrix Σ. Representation (1) is only necessary for U to generate a strongly continuouscontraction semigroup. To produce sufficient conditions requires additional restrictions.

The process X can be written as:

dXt = dXct +

∫

Rn

zζ(z, dt) (2)

where Xc is the continuous part of X and ζ = ζ(ω, ·, ·) is the random counting measure,that is for each ω, ζ(ω, b, [0, t]) gives the total number of jumps in [0, t] of a size in b, in therealization ω.

In general associated Markov stochastic process X may have an infinite number of smalljumps in any time interval. For what follows we will assume that this process has a finitenumber of jumps over a finite time interval, what rules out most Levy processes, but greatlysimplify the notation. In this case, the measure η(dy|x)dt is the compensator of the measureζ that is the (unique) predictable random measure, such that for each predictable stochasticfunction f(ω, x, t), the process

∫ t

0

∫

Rn

f(ω, y, s)ζ(ω, dy, ds)−∫ t

0

∫

Rn

f(ω, y, s)η[dy|Xs(ω)]ds

is a martingale. The measure η encodes both a jump intensity and a distribution given thata jump occurs.

We may also construct drift coefficient and diffusion matrix by performing Martingaleconstructions for the continuous part of the process familiar from Ito’s lemma. The driftcoefficient is a composite of two objects that scale the first derivative of test functions inC∞

K :

µ(x)−∫

1O(x)(y − x) · ∂φ(x)

∂xη(dy|x).

The drift construction will be independent of the original choice of set O. The diffusionmatrix is given by Σ.

Given an underlying Markov process {Xt : t ≥ 0}, we now explore ways of studyingvaluation and stochastic growth. This requires that we build processes for stochastic discountfactors and cash flow growth from the underlying Markov process. We do this by usingbuilding blocks processes that are additive and multiplicative functionals. In what followswe define formally a functional, an additive functional and a multiplicative functional anddiscuss their properties.

3.2 Additive Functionals

A functional is a stochastic process constructed from the original Markov process:

5

Definition 3.2. A real-valued process {At : t ≥ 0} is called a functional if it is adapted(At is Ft measurable for all t) and is right continuous conditioned on X0 = x for almosteverywhere for each x in D0.

Let θ denote the shift operator. Using this notation, we define Au(θt) to be the corre-sponding function of the X process shifted forward t time periods. Since Au is constructedfrom the Markov process {Xt : t ≥ 0} between dates zero and u, Au(θt) depends only theprocess between dates t and date t + u.

Definition 3.3. A functional {At : t ≥ 0} is additive if A0 = 0, it is adapted, and At+u =Au(θt) + At, for each nonnegative t and u.

While the joint process {(Xt, At) : t ≥ 0} is Markov, by construction the additive func-tional does not Granger cause the original Markov process. Instead it is constructed fromthat process. No additional information about the future values of Xt are revealed by currentand past value of At. When {Xt : t ≥ 0} is restricted to be stationary, an additive functionalhas stationary increments.

Example 3.3. Let β be a Borel measurable function on D0 and construct:

At =

∫ t

0

β(Xu)du

where∫ t

0β(Xu)du < ∞ with probability one for each t.

Example 3.4. Form:

At =

∫ t

0

γ(Xu) · dBu

where∫ t

0|γ(Xu)|2du is finite with probability one for each t and {Bt : t ≥ 0} a vector-valued

Brownian motion.

Example 3.5. Form:

At.=

∑0≤u≤t

κ(Xu, Xu−)

where κ : D0 ×D0 → R, κ(x, x) = 0.

Example 3.6. Sums of additive functionals are additive functionals. We may thus useexamples 3.3, 3.4 and 3.5 as building blocks in a parameterization of additive functionals.This parameterization uses a triple (β, γ, κ) that satisfies:

a) β : D0 → R and∫ t

0β(Xu)du < ∞ for every positive t;

b) γ : D0 → Rm and∫ t

0|γ(Xu)|2du < ∞ for every positive t;

c) κ : D0 ×D0 → R, κ(x, x) = 0 for all x ∈ D0.

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Form:

At =

∫ t

0

β(Xu)du +

∫ t

0

γ(Xu) · dBu +∑

0≤u≤t

κ(Xu, Xu−),

which is an additive process. This additive process is a semi-martingale.

Another useful set of examples is given by :

Example 3.7. Suppose that {Xt : t ≥ 0} is a standard Brownian motion, b a Borelian inR, and define the occupation time of b up to time t as

At.=

∫ t

0

1{Xu∈b)}du.

At is an additive functional. As a consequence, the local time at a point r defined as

Lat

.= lim

ε↓01

2ε

∫ t

0

1{Xu∈(r−ε,r+ε)}du,

is also an additive functional.

3.3 Multiplicative Functionals and Semigroups

Definition 3.4. The functional {Mt : t ≥ 0} is multiplicative if M0 = 1, and Mt+u =Mu(θt)Mt.

Products of multiplicative functionals are multiplicative functionals. Exponentials ofadditive functionals are strictly positive multiplicative functionals. Conversely, the logarithmof strictly positive multiplicative functional is an additive functional.

Given a multiplicative functional {Mt : t ≥ 0}, our aim to to construction a family ofoperators using the formula:

Mtψ(x) = E [Mtψ(Xt)|x0 = x] , (3)

and to establish that this family is a semigroup. With this in mind, let (L, ‖ · ‖) denotethe Banach space of (essentially) bounded functions of a Markov state under the sup norm.Among other things, the following assumption guarantees that (3) defines a bounded operatoron L for every t.

Assumption 3.1. The multiplicative functional {Mt : t ≥ 0} has finite first moments foreach t and

limt↓0

supx∈D0

E ( |Mt − 1| |X0 = x) = 0. (4)

Proposition 3.1. Suppose {Mt : t ≥ 0} is a multiplicative functional satisfying Assumption3.1. Then

Mtψ(x) = E [Mtψ(Xt)|X0 = x] .

is a strongly continuous semigroup on the space of bounded measurable functions.

7

Proof. Suppose that ψ is a bounded measurable function. Then

Mt+uψ(x) = E (E [Mt+uψ(Xt+u)|Ft] |X0 = x)= E [E (MtMu(θt)ψ[Xu(θt)]|Ft) |X0 = x]= E [MtE [Mu(θt)ψ[Xu(θt)]|X0(θt)] |X0 = x]= E [MtMuψ(xt)|X0 = x]= MtMuψ(x),

which establishes the semigroup property. Strong continuity follows directly from Assump-tion 3.1.

Since the logarithm of a strictly positive multiplicative process is an additive processwe will consider parameterized versions of strictly positive multiplicative processes by pa-rameterizing the corresponding additive process. For instance, if M = exp(A) when A isparameterized by (β, γ, κ) as in example 3.6 above, we will say that the multiplicative processM is parameterized by (β, γ, κ). Notice that Ito’s lemma guarantees that:

dMt

Mt−=

[β(Xt−) +

|γ(Xt−)|22

]dt + γ(Xt−)dBt + exp [κ(Xt, Xt−)]− 1.

4 Pricing Semigroup

For a fixed time interval t and any ψ in L, consider a state-contingent payoff ψ(Xt). Apricing operator St applied to ψ gives the time zero price of this state-contingent payoff.This price is a function of the date zero state. We construct such an operator for everyhorizon t, giving us a family of operators that are naturally restricted to be a semigroup.

Formally, we build the pricing semigroup using a stochastic discount factor process {St :t ≥ 0} via:

Stψ(x) = E [Stψ(Xt)|x0 = x] .

The stochastic discount factor process is restricted to be a strictly positive multiplicativefunctional satisfying Assumption 3.1. With intermediate trading dates, the time t+ s payoffψ(Xt+s) can purchased at date zero or alternatively the claim could be purchased at datet at the price Ssψ(Xt) and in turn this time t claim can be purchased at time zero. Thesemigroup property captures the notion that the date zero prices of ψ(Xt+s) and Ssψ(Xt)must coincide. Thus the semigroup property is an iterated value property that connectspricing over different time intervals. It is a version of the Law of One Price as it applies overtime in a frictionless market model with intermediate trading dates.

Although we use the stochastic discount factor to price claims that are functions of theMarkov state at a future date, once we choose a stochastic discount factor we can also priceclaims which are functions of the whole history up to a certain date.

8

5 Valuation Functionals and Returns

A valuation process has the interpretation that if the future value of the process is thepayout, the current value is the price of that payout. For instance a valuation process couldbe the result of continually reinvesting dividends in a primitive asset. Equivalently, it can beconstructed by continually compounding realized returns to an investment. To characterizelocal pricing, we use valuation processes that are multiplicative functionals. Recall thatthe products of two multiplicative functionals is a multiplicative functional. The followingdefinition is motivated by the connection between the absence of arbitrage and the martingaleproperties of properly normalized prices.

Definition 5.1. A valuation functional {Vt : t ≥ 0} is a multiplicative functional suchthat the product functional {VtSt : t ≥ 0} is a martingale.

Provided that {Vt : t ≥ 0} is strictly positive, a valuation functional can be transformedto gross returns over any horizon u by forming ratios:

Rt,t+u =Vt+u

Vt−

Thus increments in the value functional scaled by the current value gives an instantaneousnet return. The martingale property of the product {VtSt : t ≥ 0} gives a local pricingrestriction for returns.

To deduce a convenient and familiar risk return relation, consider the multiplicativefunctional Mt = VtSt where {Vt : t ≥ 0} is parameterized by (βv, γv, κv) and {St : t ≥ 0} isparameterized by (µs, γs, κs). In particular, the implied return evolution is:

dVt

Vt−=

[β(Xt−) +

|γv(Xt−)|22

]dt + γv(Xt−)dBt + exp [κv(Xt, Xt−)]− 1.

Thus the expected net rate of return is:

εv.= βv +

|γv|22

+

∫(exp [κv(y, ·)]− 1) η(dy, ·).

Since both {Vt : t ≥ 0} and {St : t ≥ 0} are exponentials of additive processes, theirproduct is the exponential of an additive process and is parameterized by:

β = βv + βs

γ = γv + γs

κ = κv + κs

A necessary condition for {Mt : t ≥ 0} to be a martingale,

β +|γ|22

+

∫(exp [κ(y, ·)]− 1) η(dy, ·) = 0. (5)

9

Proposition 5.1. A valuation process parameterized by (µv, γv, κv) satisfies the pricing re-striction:

βv + βs = −|γv + γs|22

−∫

(exp [κv(y, ·) + κs(y, ·)]− 1) η(dy, ·). (6)

Proof. This follows from the definition of a valuation functional and the martingale restric-tion (5).

This restriction is essentially local and determines the instantaneous risk-return relation.The parameters (σv, κv) determine the Brownian and jump risk exposure. The followingcorollary gives the required local mean rate of return:

Corollary 5.1. The required mean rate of return for the risk exposure (γv, κv) is:

εv = −βs − γv · γs − |γs|22

−∫

(exp [κv(y, ·) + κs(y, ·)]− exp [κv(y, ·)]) η(dy, ·)

The vector −γs contains the factor risk prices for the Brownian motion components. Thefunction κs is used to price exposure to jump risk. This local relation is familiar from theextensive literature on continuous-time asset pricing.

Remark 5.1. One specific example of a valuation functional is Vt = 1/St. The product of Vt

and St is constant and hence a degenerate martingale. This valuation functional is the onefeatured in the analyses of Bansal and Lehmann (1997) and Alvarez and Jermann (2004).They study the return that maximizes the expected growth rate or expected logarithm. Amongthe valuation functionals, it is straightforward to verify that this one is the one that maximizesE (log Vt+ε − log Vt) independent of the chosen horizon.2 This valuation functional obtainedthrough optimization is used to reveal information about the underlying stochastic discountfactor.

6 Reference Growth Functional

The pricing semigroup we have thus far constructed only assigns prices to payoffs of formψ(Xt). When {Xt : t ≥ 0} is stationary, this specification rules out stochastic growth. Wenow extend the analysis to include payoffs with stochastic growth components by introducinga reference stochastic growth process: {Gt : t ≥ 0}.Definition 6.1. A reference growth functional {Gt : t ≥ 0} with growth rate δ is amultiplicative functional for which {exp(−δt)Gt : t ≥ 0} is a martingale for some δ > 0.

We consider cash flows of the form:

Dt = Gtψ(Xt)D0

2In econometric analyses, the optimization is typically performed over an incomplete set of assets. As aconsequence, St is not revealed but bounds a lower bound is computed on the maximized objective.

10

for some initial condition D0. Think of ψ(Xt) as the stationary component of the cash flow.While the reference growth process is always positive, and its expected growth rate is theconstant δ, the realized cash flow can be negative and the (conditional) expected growth rateneed not be constant because of the contribution of the transient component ψ(Xt). Thiscomponent can be negative and it can be predictable.

To price payoffs with stochastic growth components, we are led to study an alternativesemigroup:

Qtψ(x) = E [GtStψ(xt)|x0 = x]

Then the date zero price assigned to Dt is D0Qtψ(X0). Different reference growth processesgive rise to alternative semigroups.

When the growth process is degenerate and equal to unity, then this semigroup is iden-tical to one constructed previously in section 4. This semigroup is useful in the study thevaluation of stationary cash flows including discount bonds and the term structure of inter-est. It supports local pricing and generalizations of the analyses of Backus and Zin (1994)and Alvarez and Jermann (2004) that use fixed income securities to make inferences abouteconomic fundamentals.

In our investigation of long run valuation, the study of this particular semigroup cor-responds to one with no long run risk exposure. By introducing stochastic growth in theform of a reference growth process {Gt : t ≥ 0} we explore the consequence to valuation ofexposure to long run risk.

Since the product of multiplicative functionals is a multiplicative functional, in whatfollows we are led to study the long run behavior of a general multiplicative semigroup:

Mtψ(x) = E [Mtψ(Xt)|X0 = x]

for some strictly positive multiplicative functional {Mt : t ≥ 0}. This functional could beformed by multiplying a stochastic discount factor St and a growth factor Gt, both of whichare multiplicative functionals.

7 A Peek at Things to Come

It will turn out that for cash flows with common reference growth processes, their valuationover long horizons will be approximated by a single dominant component. Thus for the tailcontributions to the cash flows, the contributions that come many periods into the futurewill be approximated by a single valuation or pricing factor. While this pricing factor is statedependent, the implied price to cash flow ratio will be constant. Its magnitude reflects anexpected growth rate and a risk adjustment, both of which do not depend on the transientcontribution of the cash flow.

By changing the risk exposure of the reference growth functional we change the dominantvaluation factor and we change the risk adjustment. Analogous to the instantaneous risk-return relation we just deduced, we will characterize formally a long-run risk return relationengineered by changing the growth rate risk exposure. To facilitate this construction, we will

11

associate a unique valuation functional with the reference growth functional that capturesholding period returns to cash flows far into future.

There is an alternative approach to characterization long run risk. As we have seenalternative valuation processes reflect alternative risk exposures. Given a valuation functional{Vt : t ≥ 0}, we show under suitable assumptions how to construct a corresponding referencegrowth functional {Gt : t ≥ 0} with a long run expected growth rate δ. Thus δ is the long runcounterpart to the instantaneous expected rate of return. To produce this decomposition,we are led to study semigroup {Mt : t ≥ 0} constructed using Mt = Vt.

8 Principal Eigenfunctions

The semigroup {Mt : t ≥ 0} can typically be extended from the space of bounded functionsunder the sup-norm to other Lp spaces by applying the formula:

Mtψ(x) = E [Mtψ(Xt)|X0 = x] .

We use a measure ς to define the Lp space. The choices of p and ς will typically be related tothe choice of multiplicative functional and depend on the appication. Abusing notation, letL denote the new domain of the semigroup and ‖ · ‖ its norm. We will continue to use thenotation {Mt : t ≥ 0} to denote the semigroup, even though it is now defined on a largerspace.

The semigroup always has a limiting spectral radius given by:

limt→∞

[sup‖ψ‖≤1

‖Mtψ(x)‖]1/t

= λ < ∞. (7)

Provided that λ 6= 0, for any ψ ∈ L,

lim supt→∞

exp(ρt)‖Mtψ‖ ≤ ‖ψ‖

where ρ = log λ. In what follows we seek a more refined analysis.We assume the underlying Markov process is recurrent and restrict the multiplicative

functional.

Assumption 8.1. There exists a t such that for any bounded nonnegative function ψ 6= 0,E[ψ(Xt)|X0 = x] > 0 for all x.

Assumption 8.2. The multiplicative functional {Mt : t ≥ 0} is strictly positive with proba-bility one.

Initially, we assume the existence of a positive eigenfunctions.

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Assumption 8.3. There exists a nonnegative solution φ in L to the principal eigenfunctionproblem, that is:

Mtφ = exp(ρt)φ

and a solution φ∗ ∈ L∗ to the dual problem, that is:

M∗t φ∗ = exp(ρt)φ∗

for all t > 0.

Below we give sufficient conditions for this assumption to hold.Given the existence, it follows that the eigenfunctions are the same for all t and that

the eigenvalues can be parameterized as exp(ρt) for some t. Moreover, this eigenfunction isalso an eigenfunction for the generator with eigenvalue ρ. In light of assumption 8.1, theeigenfunction φ is strictly positive. That is φ(x) > 0 for all x.

8.1 Uniqueness

The solution to the principal eigenfunction problem is unique up to scale. This is a standardresult from the Frobenius-Krejn theory of positive operators.

Proposition 8.1. Suppose that Assumptions 8.1, 8.2 and 8.3 are satisfied. Then the solutionφ to the principal eigenvalue problem is unique up to scale.

Proof. This is an application of Theorem 11.1 of Krasnoselsk’ij et al. (1980) under his con-dition e). The cone of an Lp is reproducing and minihedral required for their analysis. Forsufficiently large t, Assumption 8.1 and the strict positivity of the multiplicative functionalimply that Mt is irreducible. Assumption 8.3 assures the existence of a solution to the dualof the principal eigenvalue problem.

8.2 Martingales

The principal eigenvalue and eigenfunction satisfy the equation:

E [Mtφ(Xt)|X0] = exp(ρt)φ(X0)

We use these objects to re-scale the multiplicative functional:

M̂t = exp(−ρt)Mtφ(Xt)

φ(X0), (8)

which, by construction, is now a multiplicative martingale. This martingale has three appli-cations that are germane for our analysis. They are pursued in the subsections that follow.

13

Remark 8.1. Recall that we typically form multiplicative functionals by exponentiating ad-ditive functionals. There are well known martingale decompositions of additive functionalswith stationary increments used in deducing central limit approximation and in character-izing the role of permanent shocks in time series. The nonlinear, continuous time Markovversion of such a decomposition is:

At = ωt + mt − υ(Xt) + υ(X0)

where {mt : t ≥ 0} is a martingale with stationary increments (see Bharttacharya (1982)and Hansen and Scheinkman (1995)). Exponentiating this decomposition is of a similartype except that the exponential of a martingale is not a martingale. When the martingaleincrements are constant functions of Brownian increments, then exponential adjustment hassimple consequences.3 In particular, the exponential adjustment is offset by changing ω.With state dependent volatility in the martingale approximation, however, there is no longera direct link between the additive and multiplicative decompositions.

8.2.1 Constructing a Reference Growth Functional from a Valuation Functional

Previously in section 6 we introduced a reference growth functional that was restricted to bea martingale scaled by an exponential function of time. Given a cash flow process that is amultiplicative functional, we find the principal eigenfunction of the corresponding semigroupand construct the martingale as in (8). The reference growth functional is then given by:

Gt = M̂t exp(ρt)

and has growth rate ρ. Thus the original cash flow and the reference growth process arerelated by:

Mt = Gtφ(X0)

φ(Xt),

and they will share growth properties provided that {Xt : t ≥ 0} is stationary.One type of cash flow process that is of particular is a valuation functional {Vt : t ≥ 0}.

Form a semigroup using Mt = Vt. Then

Gt = Vtφ(Xt)

φ(X0)

is a reference growth process with growth rate % where φ is the principal eigenfunction andexp(%t) is the corresponding eigenvalue for interval t. As we will see, the dominant eigenvalue% will be the expected growth rate of the implied return in the long run. This approach iscommonly done in the study of long horizon returns.

To obtain one version of a long term risk return frontier, we change the risk exposure ofthe valuation functional subject to restriction (6). This gives a family of valuation functionals

3This is the case studied by Hansen et al. (2005).

14

that are compatible with a single stochastic discount factor. We may then find a referencegrowth functional and corresponding growth rate associated with each of these valuationfunctionals.

In what follows we consider a reverse approach.

8.2.2 Using a Reference Growth Functional to Construct a Valuation Functional

Let {Gt : t ≥ 0} be a reference growth functional with growth rate δ. Form a semigroupusing Mt = GtSt where {St : t ≥ 0} is a stochastic discount factor functional. Then

Vt = exp(−ρt)Gtφ(Xt)

φ(X0)

is a valuation functional where φ is the principal eigenfunction and exp(ρt) is the principaleigenvalue for interval t, associated with the semigroup Mt. Using an approximation argu-ment that we will turn to next, the returns encoded in this valuation functional have thefollowing interpretation as approximators. Consider a hypothetical security with a payoffgiven by this cash flow starting at some future date. When that date is sufficiently far intothe future, the transient component does not contribute much to the return. Instead thereturn is approximately the ratio of the valuation functional at the respective dates con-structed from the principal eigenvalue and function. The corresponding long run expectedrate of return is given by δ − ρ.

By changing the reference growth functional including its risk exposure, we trace outa long run risk return tradeoff. Hansen et al. (2005) use this apparatus to produce sucha tradeoff using empirical inputs in a discrete-time, log linear environment. The formu-lation developed here allows for extensions to include nonlinearity in conditional means,hetoroskedasticity and large shocks modeled as jump risk.

8.2.3 Constructing an Alternative Markov Process

To establish the long run dominance of the principal eigenvalue and function, we show thatthe multiplicative martingale can be used to build an alternative family of distorted Markovtransition operators:

M̂tψ(x) = E[M̂tψ(Xt)|X0 = x

].

In what follows we call the resulting semigroup the principal eigenfunction semigroup. Thissemigroup is a well defined semigroup at least on the space L∞. As we will see, this alternativesemigroup gives us a convenient way to characterize formally the sense in which a principaleigenfunction is also a dominant eigenfunction.

The Markov process that has the semigroup M̂t as its semigroup of conditional expec-tations can be initialized to be stationary, and we show this by using the adjoint problem tobuild the required stationary density. The adjoint operator has a convenient representation,which can be seen by changing the date of the conditioning information:

M∗t ψ

∗(x) = E [Mtψ∗(X0)|Xt = x]

15

for ψ∗ in the domain of M∗t . Recall that a function φ∗ in L∗ solves the dual eigenvalue

problem if:M∗

t φ∗ = exp(ρt)φ∗.

ThenE

[M̂tφ(X0)φ

∗(X0)|Xt = x]

= φ(x)φ∗(x),

implying that a stationary density for the distorted Markov process is proportional to theproduct of the two eigenfunctions multiplied by the original stationary density. Thus thestationary density for the distorted process is:

dς̂.= φ∗φdς,

once we have normalized the eigenfunction φ∗ so that:∫

φ∗φdς = 1.

It follows from Nelson (1958) that the M̂t can be extended to L̂p for any p ≥ 1 constructedusing the measure dς̂.

Assumption 8.4. For some function V ≥ 1

limt→∞

sup0≤|ψ|≤V

∣∣∣∣M̂tψ(x)−∫

ψφφ∗dς

∣∣∣∣ = 0

pointwise in x.

When the function V = 1, this assumption follows from Assumptions 8.1, 8.2 and 8.3 as animplication of Orey’s Theorem [REFERENCE]. It is sometimes useful however to use alter-native functions V in the proposition that follows and sufficient conditions for Assumption8.4 to hold for more general functions are given in a variety of places including Meyne andTweedie (1993). We use this approximation result for continuous time Markov processesto show formally that the principal eigenfunction is in fact the dominant eigenfunction formultiplicative semigroups.

Proposition 8.2. Under Assumption 8.4, for any ψ such that |ψ| ≤ φV

limt→∞

exp(−ρt)Mtψ(x) = φ(x)

∫ψφ∗dς.

for all x.

Proof. For any ψ, φ∗|ψ| ≤ V . Under Assumption 8.4,

limt→∞

exp(−ρt)Mtψ(x) = φ(x) limt→∞

M̂t

(ψ

φ

)(x) = φ(x)

∫ψφ∗dς.

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The limit in this proposition characterizes the rate of growth or decline of the multiplicativesemigroup. After adjusting for this growth by scaling by exp(−ρt), the limit as a function ofthe state x is proportional to the dominant eigenfunction provided that

∫ψφ∗dς is different

from zero. Thus it provides a refined characterization of the limit. This is true, independentof the choice of ψ given this proviso. For instance, when the multiplicative semigroup isused to depict values, its limiting behavior is state dependent as captured by the principaleigenfunction, but the state dependency is invariant to the choice of ψ. Roughly speaking,ψ is a transient component that is dominated in the long run.

9 Existence

So far, we have shown when a principal eigenfunction exists. We now study conditions forexistence. One approach is to work with a member of the semigroup for a single horizon,say t = 1. We adopt this approach here and give some give some sufficient conditions forAssumption 8.3.

Assumption 9.1. The limiting spectral radius λ, defined in (7), is strictly positive.

Assumption 9.2. M1 is compact.

The following generalization of the Frobenius-Perron theorem for non-negative matrices,known as the Krein-Rutman theorem, holds:

Proposition 9.1. If Assumptions 9.1 and 9.2 are satisfied, there exists a positive φ in Lwith M1φ = λφ and a positive φ∗ ∈ L∗ with M∗

1φ∗ = λφ∗.

(See Krein and Rutman (1948) and Bonsall (1958)).

10 Generator a Multiplicative Semigroup

When we first introduced a multiplicative semigroup in section 3.3 we used the space ofbounded functions as its domain. For many purposes, this space is too small, and it isadvantageous to have more flexibility when studying the local behavior of the semigroup.Next consider an alternative approach based on taking limits as we shrink the time intervalto zero. This approach is more delicate mathematically, but it gives rise to equations thatcan be solve for the principal eigenfunction. To make this approach operational we need todefine the full and extended generators of a multiplicative semigroup.

For a given multiplicative functional {Mt : t ≥ 0}, let L̃ be the space of all real valuedfunctions ψ such that for every t ≥ 0,

sup0≤s<t

E[Ms|ψ(Xs)|] < ∞.

Consider a semigroupMtψ(x) = E [Msψ(Xs)|X0 = x] .

17

To make the construction of a generator operational, we compute a derivative with respectto time

χ(x).= lim

t↓0Mtψ(x)− ψ(x)

t

pointwise in x. Formally, we use the operator counterpart of the relation between an integraland its derivative to define what is referred to as the full generator.

Definition 10.1. The full generator of the semi-group Mt is the subset of functions

A =

{(ψ, χ) : Mtψ − ψ =

∫ t

0

Msχds

}.

Proposition 10.1. If (ψ, χ) ∈ A then Nt = Mtψ(Xt)−∫ t

0Msχ(Xs)ds is a martingale with

respect to the filtration {Ft : t ≥ 0}.Proof.

E[Mt+uψ(Xt+u)|Ft]− E

[∫ t+u

0

Msχ(Xs)ds|Ft

]=

MtE [Mu(θt)ψ(Xt+u)|Ft]−∫ t

0

Msχ(Xs)ds− E

[∫ t+u

t

MtMs−t(θt)χ(Xs)ds|Ft

]=

= Mt

[Muψ(Xt)−

∫ u

0

Msχ(Xs+t)

]−

∫ t

0

Msχ(Xs)ds = Nt,

since (ψ, χ) ∈ A.

This last proposition leads to the definition of an extended generator.

Definition 10.2. A Borel function ψ belongs to the domain of the extended generator ofthe semi-group Mt if there exists a Borel function χ such that Nt = Mtψ(Xt)−

∫ t

0Msχ(Xs)ds

is a martingale with respect to the filtration {Ft : t ≥ 0}.If instead we define the semigroup {Mt : t ≥ 0} on a smaller Banach space such as the

construction in section 3.3, we may construct the infinitesimal generator as

Aψ = limt↓0

Mtψ − ψ

t,

where the limit is defined in the Banach space norm. This domain D(A) of A is the subset ofBanach space for which this limit exists. As before we assume thatMt is strongly continuouson this Banach space. When ψ is in D(A), the pair (ψ,Aψ) is in the full generator A. Thusthe full generator gives a more general construction that is consistent with the semigroupgenerator we discussed previously. In turn the extended generator extends the full generator.

Recall our earlier parameterization of an additive process in terms of the triple (β, γ, κ).The exponential of this gives a parameterization of a multiplicative process. This multi-plicative process implies a semigroup with a corresponding generator. We now show how

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to go from the generator of the original Markov process to the generator of the semigroupassociated with the multiplicative process. Factor the diffusion matrix Σ = σσ′ where σhas the same number of columns as the rank of Σ. The matrix Σ is restricted to have aconstant rank on the interior of the state space. This parameterization of the multiplicativeprocess implies an altered parameterization of the generator. In this construction we use thefull generator in order to avoid the need to construct a possibly different Banach space foralternative multiplicative semigroups.

The formulas below use the parameterization for the multiplicative process to transformgenerator of the Markov process into the generator of the multiplicative semigroup:

a) α(x) → α(x) + β(x) + |γ(x)|22

+∫

(exp [κ(y, x)]− 1) η(dy, x);

b) µ(x) → µ(x) + σ(x)γ(x);

c) Σ(x) → Σ(x);

d) η(dy|x) → exp [κ(y, x)] η(dy|x).

As stated earlier, typically the parameter α is zero.There are a variety of direct applications of this analysis. In the case of the pricing

semigroup introduced in section 4, the generator encodes the local prices reflected in the localrisk return tradeoff of Proposition 5.1. The modified version of α gives the instantaneousversion of a risk free rate. In the absence of jump risk, the increment to the drift gives thefactor risk prices. The function κ shows us how to value jump risk in small increments intime.

In a further application of this decomposition, Anderson et al. (2003) use this decom-position to characterize the relation among four alternative semigroups, each of which isassociated with an alternative multiplicative process. Anderson et al. (2003) feature modelsof robust decision making. In addition to the generator for the original Markov process, asecond generator depicts the worst case Markov process used to support the robust equilib-rium. There is a third generator of an equilibrium pricing semigroup, and a fourth generatorof a semigroup use to measure the statistical discrepancy between the original model andthe worst-case Markov model.

As we have have already argued, a long run perspective on valuation is captured byprincipal eigenvalues and eigenfunctions that dominate as the valuation horizon is extended.The full generator gives an equation to be solved, which we write as:

Aφ = ρφ

for a positive function φ. An equation such as this suggests a different approach to thestudy of existence. We may solve an equation implied by the generator instead of an equa-tion implied by the one period operator. This latter approach is more operational becausecontinuous-time modeling often represents equilibrium outcomes in terms of the implied localevolution and hence the implied A.

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In the case of diffusions, Chen et al. (2005) give some primitive conditions for the caseof multivariate stationary diffusions that imply complete eigenvalue decompositions, andhence as a special case the existence of a solution the principal eigenvalue problem.4 Theirconditions use a symmetrized version of the generator. Analyses that focus on directly theprincipal eigenvalue problem in general domains are given in Berestycki et al. (1994) andBereskycki et al. (2005).

11 Examples

11.1 Log-linear example

Consider an environment with stationary linear dynamics:

dXt = AXtdt + CdBt

where {Bt : t ≥ 0} is a multivariate standard Brownian motion. Introduce a stochasticdiscount factor of the form

St = exp

(∫ t

0

Θ′dBs +

∫ t

0

Λ′Xsds− λt

),

a reference growth process of the form:

Gt = exp

[∫ t

0

(Γ′dBs − 1

2Γ′Γ

)ds + εt

].

The generator for the corresponding semigroup is: The generator for the valuation semigroup:

• level coefficient

−λ + ε− 1

2Γ′Γ +

1

2(Θ + Γ)′(Θ + Γ) + Λ′x

• first derivative coefficientAx + CΘ + CΓ

• second derivative coefficientCC ′

The principal eigenfunction is the exponential of the linear function of the state vector:

φ(x) = exp(Q · x).

To compute Q we must solve:Λ + A′Q = 0,

4Chen et al. (2005) presume that the A operator does not have a level term, but it is straightforward toadd such a term to their framework.

20

and the principal eigenvalue is given by:

ρ = −λ + ε +|Θ|22

+ Θ · Γ +1

2trace(CC ′QQ′) + Q′CΘ + Q′CΓ

The long run counterpart to the price of risk is:

Θ′ − Λ′A−1C,

which is also the vector of permanent responses of the stochastic discount factor to theBrownian increment. These are growth prices to exposure to Brownian motion risk. Theyare the long run counterpart to local risk prices given by the vector Θ′.

Hansen et al. (2005) give a structural interpretation of this model and discuss measure-ments using discrete time vector autoregressive methods. This example is special because itprecludes the role of volatility. Next we consider an example with volatility included.

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