# State-Dependent Utilities and Incomplete Markets

Authors: Jaime A. Londoño
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The problem of optimal consumption and investment
for an agent that does not influence the market is solved. The optimization
criteria are based on a state-dependent utility functional as proposed
in Londoño (2009). The proposed solution is given in any market without state-tame arbitrage opportunities, includes several utilities structures, and includesincomplete
markets where there are multiple state variables. The solutions obtained
for optimal wealths consumptions, and portfolios are explicit and
easily computable; the main condition for the result to hold is that the income
process of each agent is hedgeable, requiring a natural condition on
employer and employee to agree on a contract whose risk can be managed
by both parties. In this paper we also developed a theory of markets
when the processes are generalization of Brownian flows on manifolds,
since this framework shows to be the natural one whenever the problem of
intertemporal equilibrium is addressed.


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## Body

## 1. Introduction

The problem of optimal consumption and investment for a “small investor” whose actions do not influence market prices incomplete markets and where consumers have dependent utility structures has been studied in Londoño [1]. The modern treatment of this problem when the asset prices follow Itô processes started with the seminal works of Merton [2, 3]. Using a “martingale” approach, Cox and Huang [4] and Karatzas et al. [5] solved the problem in more general settings in the case of complete markets. Analytical and numerical problems with those earlier solutions and lack of agreement with empirical data motivated alternative treatments; see Londoño [1] for some literature related with problems associated with the standard models as well as some reviews of other approaches.In incomplete markets, there are even more inconveniences associated with the theory of optimal consumption and investment. General results have been derived in Karatzas and Žitković [6], Hugonnier and Kramkov [7], and Cvitanić et al. [8]. The solutions obtained by them are very limited, and almost nontrivial cases have been solved explicitly. We notice that even though the utilities structures studied by some of them allow state dependence, they are not able to handle the case presented in this paper since they ask for bounded (with bounds that depend on time and value of consumption) utility random fields that do not include the ones considered here; see Karatzas and Žitković [6, Definition 3.1 and Proposition 3.5. (item 1)]. In the case presented in this paper the utility random fields are usually considered as unbounded (in the state variable) when the time and the consumption are fixed; see Remark 8. In case of hedgeable (insurable) random endowments and incomplete markets the solution is well known in the case of Markovian markets with non state variables. For instance Merton [3] stated that, in computing the optimal decision rules, the individual capitalizes the lifetime flow of wage income at the market (risk-free) rate of interest and then treats the capitalized value as an addition to the current stock of wealth. Similar results are obtained in more general Markovian markets, as is discussed in Karatzas and Shreve [9]. This result is even true for semimartingale complete markets as is pointed out in Karatzas and Žitković [6].Whenever state variables are introduced, the solutions provided to complete markets are no longer available even in the presence of hedgeable income, or even with no income. Computable solutions are not known in general settings, since hedgeability of the income structure does not necessarily allow for the optimal portfolio to be hedgeable in the standard models (see Karatzas and shreve [9, Sections 3.6 and 4.4]). However, current models of equilibrium allow for the existence of state variables that model the dynamics of the relevant variables of the economy as in Merton [10], Breeden [11], Cox et al. [12], or Londoño [13] to cite just a few.The main result of this paper is the solution to the problem of optimal consumption and investment for an agent that does not influence the market. It is assumed that the utility maximization criteria used are based on a state-dependent utility functional as proposed in Londoño [1]. For the optimization problem of this paper, it is assumed that consumers are endowed with an initial capital and a hedgeable random endowment where the underlying market is not necessarily assumed to be complete.Here, we extend further the approach presented in Londoño [1]. In this model utilities reflect the level of consumption satisfaction of flows of cash in future times as they are valued by the market when the economic agents are making their consumption and investment decisions. The utilities used in this paper were introduced in Londoño [1] and are equivalent to state-dependent utilities in standard settings, where dependence on the state is through the state price density process (see (7) and Remark 8). The main assumption of the theorem, besides the interpretation of the utility functionals, is that the income of the agent should be hedgeable; in plain English it is required that the employer and employee agree on a labor contract that allows each party to hedge any risk associated with the this contract, and in this way any economic agent would be able to cover the compromises of this contract in exchange for a fair price. In the context of this paper it is not needed to be able to cover all the financial liabilities in a given economy (for instance it is not needed that any “reasonable” derivative could be hedge).The solutions of the optimal consumption and portfolio problem are obtained in a very general setting which includes several functional forms for utilities and considers general restrictions on allowable wealth that are used in the current literature. We obtainedsimple and computable solutions that are optimal consumption and investment strategies in all studied cases. In our model it is always true that when the endowment is hedgeable, the problem becomes equivalent to one where the entire endowment process is replaced by its present value, in the form of an augmented initial wealth; see Corollary 9.The theoretical framework proposed in this paper is one of stochastic flows on manifolds; this is an extension of the framework proposed in Londoño [14]. In the spirit of Merton [10], Breeden [11], and Cox et al. [12] we assume a Markovian setting for the “state variables” that includes not only the price processes but also additional variables that describe the evolution of the economy. This setting proved to be the right framework to study equilibrium problems, since equilibrium defines restrictions on the variables that make them take values on manifolds; see Londoño [13].This paper also depends on the valuation and arbitrage theory presented in Londoño [14]; this is an extension of the theory of state tameness (Londoño [15]) that provided a unified framework for valuation of financial instruments, of both European and American types, with an algebraic appealing character and economic justification. In Londoño [14, 15] the conditions presented for valuation of financial instruments of American type are the weakest possible. Additional characteristics of the framework developed in Londoño [14, 15] are weak conditions on the coefficients on the volatility matrix of the price process and a development of a theory of valuation of contingent claims with random expiration date.To the best of our knowledge this theory of arbitrage and valuation is the most general existing setting in the case of (continuous) semimartingales driven by Brownian filtrations with continuous coefficients. For a review of the state of the art on valuation and arbitrage theory we suggest the reader to look at Londoño [15] and the references therein.


## 2. The Model

First we introduce some notation which will be frequently used in this paper. Let𝔻⊂[s,T]×ℝk be a measurable set for 0≤s<T, with section 𝔻(t)={x∈ℝk∣(t,x)∈𝔻} for s≤t≤T. We assume that for each t, 𝔻(t) has a differential structure, which at this point is not necessary to specify. Examples of this differential structure are sets whose sections 𝔻(t) are the solution sets {x∣φ(t,x)=0} of a function φ:[s,T]×U→ℝp for some p<k, where U⊂ℝk is an open set, and φ is a continuous function, that is, differentiable in the spatial variable, and whose partial derivatives are continuous (also in the time variable). Moreover, in order to define a manifold structure it is customary to impose that the differential Dφ(t,·) has maximal rank for each t. Other examples are sets whose sections 𝔻(t) are integral manifolds defined by k-p (1≤p<k) continuous vector fields X1s,…,Xk-ps:[s,T]×U→ℝk, where U is an open set as above, and X1s(t,·),…,Xk-ps(t,·) are continuously differentiable vector fields that are linearly independent at each space point. It is well known that these two conditions above imply a differential structure on each of the specified subsets; see Warner [16]. In this paper it is always assumed that the degree of “smoothness” of the differential structure is sufficient for every definition to make sense. By this we mean that the transition functions, the functions defining the solution sets or the vector fields, are sufficiently differentiable.Letm be a nonnegative integer, and let 𝔻 be a set defined as above. We say that f∈Cm,δ(𝔻:ℝn) if f is a continuous function f∈C(𝔻,ℝn), and there exist an open set 𝕌 (relative to the topology of [0,T]×ℝk) with 𝔻⊂𝕌⊂[0,T]×ℝk and a continuous extension f~ of f such that f~(t,·)∈Cm,δ(𝕌(t):ℝn), where this last space is the Fréchet space of m-times continuous differentiable functions whose mth-order derivatives are δ-Hölder continuous with seminorms ∥·∥m,δ:K defined in Kunita [10, Section 3.1] with ∫sT‍∥f~(t,·)∥m,δ:Kdt<∞, where K⊂𝕌 is a compact set, 0≤δ≤1. In case m=0 (or δ=0), we denote Cm,δ(𝔻:ℝn) simply by Cδ(𝔻:ℝn) (Cm(𝔻:ℝn)), and whenever clear we denote the above spaces simply as Cm,δ, Cδ, and Cm, respectively. We denote by Cm,0+ the set ∪δ>0Cm,δ, and C0+=C0,0+. Let 𝔻′⊂[s,T]×ℝk be a measurable set with sections 𝔻′(t); we say that a function f belongs to the class Cm,δ(𝔻:𝔻′), where δ>0 or δ=0+, if f(t,·) takes values in 𝔻′(t) and f∈Cm,δ(𝔻:ℝn). We assume in the following definitions that we have a differential structure sufficiently smooth according to the space defined. We notice that the definition of Hölder continuity is made with respect to the Euclidean distance derived from the fact that these sets are subsets of an Euclidean space.Definitions of consistent processes that we review below are natural generalizations (of processes defined on embeddings) of the definitions made in Londoño [15].We assume ad-dimensional Brownian motion {W(t),ℱt;0≤t≤T} starting at 0 and defined on a complete probability space (Ω,ℱ,P), where ℱ=ℱT and {ℱt,0≤t≤T} is the P augmentation by the null sets of the natural filtration ℱtW=σ(W(s),0≤s≤t). Let (ℱs,t)={ℱs,t,0≤s≤t≤T} be the two-parameter filtration, where ℱs,t is the smallest sub-σ-field containing all null sets and σ(Ws(u)∣s≤u≤t), where Ws(u)≡W(u)-W(s). For each 0≤s≤T we also define the σ-field 𝒫s of progressive measurable sets after time s as the σ-field of sets P∈ℬ([s,T])⊗ℱs,T, and  the product σ-field, such that χP(t,ω), t≥s, is a ℱs,t progressive measurable (in t) process, where χ is the indicator function. We denote by μs the measure on 𝒫s defined by μs(P)=E∫sT‍χP(t,ω)dt.Letφs,t(x,ω), 0≤s≤t≤T, x∈𝔻(s), be a ℝn-valued random field on the probability space (Ω,ℱ,P), where 𝔻 has a differential structure sufficiently smooth for the following definitions to make sense, and let 𝔻′⊂[0,T]×ℝn be a measurable set. We say that φs,t is a continuousCm,δ(𝔻:𝔻′)-semimartingale if φs:t→φs,t(·) is a continuous random field with values in Cm,δ(𝔻∩([s,T]×ℝk):𝔻′∩([s,T]×ℝn)), that is, a continuous (ℱs,·) semimartingale process. In addition we assume that φs,t(x) can be decomposed as φs,t(x)=φs,tloc(x)+φs,tfv(x), where φs,·loc(·) is a continuous Cm,δ(𝔻∩([s,T]×ℝk):𝔻′∩([s,T]×ℝn))-local-martingale, and φs,·fv(·) is a continuous Cm,δ(𝔻∩([s,T]×ℝk):𝔻′∩([s,T]×ℝn))-process of bounded variation for each 0≤s≤T. A pair (a,b), where as,t(x,y) and bs,t(x) are measurable random fields ℱs,t-progressive measurable in t for all x,y∈𝔻(s), 0≤s≤T, is said to be the local characteristics ofφ if (as,·(x,y),bs,·(x)) is the local characteristic of φs≡φs,·(·) for any s≤T, (see Kunita [17]). In addition, a pair (σ,b), where σs,t(x) is a measurable random field, that is, (ℱs,t)-progressive measurable in t with values in the set L(ℝd:ℝn) of real-valued matrices with size n×d, and b is as above, is said to be the diffusion and drift processes ofφ if
(1)φs,tloc(x)(ω)=∫st‍σs,u(x)dWs(u)
for all x, s, t, and ω. If bs,·(·) and σs,·(·) are processes of class Cm,δ for all 0≤s≤T, we will say that φ has diffusion and drift of classCm,δ.Letφs,t(x) and ψs,t(x) be continuous C(𝔻:ℝn) and C(𝔻:𝔻) semimartingales, respectively, where in addition, it is assumed that ψ(s,s,x)=x for all x∈𝔻(s). We say that the process φ is a ψ-consistent semi-martingale process if for each 0≤s≤s′≤T there exists a set Ns,s′∈𝒫s′ with μs′(Ns,s′)=0, such that φs,t(x)=φs′,t(ψs,s′(x)) for all (t,ω)∉Ns,s′ and all x∈𝔻(s). We say that the process φ is a consistent semi-martingale process if φ is a φ-consistent process.A few words should be said about the existence of solutions of stochastic differential equations on manifolds, and although our approach is surely not the more general, we believe that it is the simplest since it does not require any technicalities of differential geometry. The simplicity of our approach is due to our definitions of functions of typeCm,δ on a manifold. Assume that b:𝔻→ℝn and σ:𝔻→L(ℝn:ℝd) are functions in Cm,δ, where m=0 and δ=1 or m≥1 and δ>0. Let φs,t(x) be the local (maximal) solution of
(2)φs,t(x)=b~(t,φs,t(x))dt+σ~(t,φs,t(x))dWs(t),φs,s(x)=x
for x∈𝕌(s), where b~ and σ~ are the extensions of b and σ to some open set 𝕌. We assume that φs,t(x) is the local solution of class Cm,ϵ for any 0≤ϵ<δ, see Kunita [17, Theorems 4.7.1 and 4.7.2]. If for any x∈𝔻(s), φs,t(x) has a nonexplosive solution with values in 𝔻, we will say that φs,t(x), x∈𝔻(s) is the solution to the stochastic differential equation on 𝔻. It is straightforward to see that this defines uniquely a process that does not depend on the extensions b~ and σ~ that are used.Next, we describe a financial market. We assume anm-dimensional Itô process Θ of two parameters with values in C2,0+(𝕂:𝕂) (for m>1), where 𝕂(s)⊂ℝ+×ℝm-1 (where ℝ+=(0,∞)) for each 0≤s≤T, and 𝕂⊂[0,T]×ℝ+×ℝm-1 is some measurable set with a differential structure as discussed above. We assume an n+m-dimensional Itô process (P,Θ) of two parameters with values in C0+(𝔻:𝔻), where 𝔻 is a measurable set with sections 𝔻(s)⊂ℝ+n×𝕂(s) for each s, where ℝ+n=(ℝ+)n and with a differential structure as discussed above. We assume that for each initial condition ϑ=(ϑ0,…,ϑm-1)⊤∈𝕂(s),
(3)dΘs,t0Θs,t0=(r(t,Θs,t)dt+∥θ(t,Θs,t)∥2)dt+∑1≤j≤d‍θj(t,Θs,t)dWsj(t),Θs,s0(ϑ)=ϑ0,
where ∥·∥ denotes the Euclidean norm, and
(4)Θs,ti=ρi(t,Θs,t)dt+∑1≤j≤d‍ϱi,j(t,Θs,t)dWsj(t)Θs,si(ϑ)=ϑi,i=1,…,m-1
for some continuous functions r,ϑj,ρi,ϱi,j for j=1,…,d of class C2,0+, and i=1,…,m-1 for which global solutions of these stochastic differential equations exit. We also assume that for each (p1,…,pn,ϑ0,…,ϑm-1)⊤=(p⊤,ϑ⊤)⊤∈𝔻(s),
(5)dPs,tiPs,ti=bi(t,Ps,t,Θs,t)dt+∑1≤j≤d‍σi,j(t,Ps,t,Θs,t)dWsj(t)Ps,si(p,ϑ)=pi,i=0,…,n,
where, for 1≤i≤n, Ps,ti=πi∘Ps,t is the price of shares outstanding for thei-stock and where πi denotes the projection on the ith-component for 1≤i≤n. We assume that σi,j, bi are continuous functions of class C2,0+ for which global solutions of the set of differential equations above exist and are unique.We point out that in a free of (state) arbitrage opportunities and (state) complete market,ϑ0/Θs,t0 is the process (with two parameters) that discounts the flow of money in every future state t, s≤t≤T to bring the value of the flow to current time s; for instance see Londoño [15] and the definition of Hs,t below.The process of bounded variationBs,t≡Bs,t(ϑ), whose evolution is given by the stochastic differential equation
(6)dBs,t=r(t,Θs,t)Bs,tdt,Bs,s=1,fors≤t≤T,
will be called the bond price process.We define thestate price density processHs,t≡Hs,t(ϑ) to be the continuous C0+(𝕂:ℝ+) process given by
(7)Hs,t=Bs,t-1Zs,t,s≤t≤T,
where Zs,t≡Zs,t(ϑ) is the process defined as
(8)Zs,t=exp{-∫st‍θ⊤(u,Θs,u)dW(u)-12∫st‍∥θ(u,Θs,u)∥2du}
for s≤t≤T, and Bs,t-1=1/Bs,t. We point out that
(9)Hs,t=ϑ0Θs,t0
easily follows the Itô’s lemma. Throughout this paper we will assume that θ(t,ϑ)∈ker⊥(σ(t,p,ϑ)), where ker⊥(σ(·)) denotes the orthogonal complement of the kernel of σ(·) and
(10)b(t,p,ϑ)+δ(t,p,ϑ)-r(t,ϑ)1n=σ(t,p,ϑ)θ(t,ϑ)
for all (p⊤,ϑ⊤)⊤∈𝔻(s) and t, where 1n⊤=(1,…,1)∈ℝn. This latter assumption implies that there are no state-tame arbitrage opportunities (see Londoño [15]). In addition to the above condition we assume in this paper that the market satisfies the following condition that we call smooth market condition. We notice that since θ(t,ϑ)∈ker⊥(σ(t,p,ϑ))=Im(σ⊤(t,p,ϑ)), the existence of a measurable function κ with the property expressed in (11) follows for any financial market, and therefore the condition below is indeed a weak condition on the smoothness of the mentioned property.Condition 1 (smooth market condition).
There exists aC2,0+-matrix-valued function κ defined on 𝔻 with the property that
(11)σ⊤(t,p,θ)κ(t,p,ϑ)=θ(t,ϑ).
We point out that we require no dependence of prices on the evolution of the functionsr and θ. We say that bs,t≡b(t,Ps,t,Θs,t) is the return process, σs,t≡σ(t,Ps,t,Θs,t) is the volatility process, δs,t≡δ(t,Ps,t,Θs,t) is the process of dividends, rs,t≡r(t,Θs,t) is the interest rate process, and θs,t≡θ(t,Θs,t) is the market price of risk process.
For a structure as above, we say thatℳ=(P,Θ,𝔻,𝕂,b,σ,δ,θ,ρ,ϱ,r,p0,ϑ0) is a financial market with terminal timeTand initial time0, feasible set of values𝔻, feasible set of state values𝕂, vector of returnsb(t,·)=(b1(t,·)…,bn(t,·))⊤, matrix of volatility coefficientsσ(t,·)=(σi,j(t,·)), vector of dividendsδ(t,·)=(δ1(t,·),…,δn(t,·))⊤, market price of riskθ(t,·)=(θ1(t,·),…,θd(t,·))⊤, interest rater(t,·), drift of the state processρ(t,·)=(ρ1(t,·),…,ρm(t,·))⊤, diffusion matrix for the state processϱ(t,·)=(ϱi,j(t,·)), vector of initial pricesp0∈ℝ+n, and initial state variables ϑ0.
Next, we review and extend some definitions from Londoño [1] that are needed to describe equilibrium.Definition 1.
Assume a measurable set𝕏⊂[0,T]×ℝn+m+1 with some differential structure as discussed in Section 2 with (nonempty) section 𝕏(s)⊂ℝ×𝔻(s) for each 0≤s≤T. Assume a family of continuous Itô’s processes
(12)X={Xs,t(x,p,ϑ);(x,p⊤,ϑ⊤)⊤∈𝕏(s),0≤s≤t≤T},
such that (X,P,Θ) is a consistent process of class C0+(𝕏:𝕏). We say that X is a wealth evolution structure; we will denote this by writing X∈𝒳(ℳ). For a detailed description of consistent processes and related ones see Londoño [1, 14]. One says that 𝕏 is a feasible set of values for(X,P,Θ).
Let(π0,π)={(πs,t0(x,p,ϑ),…,πs,tn(x,p,ϑ)); (x,p⊤,ϑ⊤)⊤∈𝕏(s),0≤s≤t≤T} be a (X,P,Θ)-consistent progressive measurable process of class C0+ with π0+(π)⊤1n=X. Assume that c={cs,t(x,p,ϑ):(x,p⊤,ϑ⊤)⊤∈𝕏(s),0≤s≤t≤T} is a nonnegative consistent (X,P,Θ) progressive measurable process of class C0+ and that Q={Qs,t(p,ϑ):(p⊤,ϑ⊤)⊤∈𝔻(s),0≤s≤t≤T} is a non-negative consistent (P,Θ) progressive measurable process of class C0+. It is assumed that
(13)Bs,t-1(ϑ)Xs,t(x,p,ϑ)=x+∫st‍Bs,u-1(ϑ)(Qs,u(p,ϑ)-cs,u(x,p,ϑ))du+∫st‍Bs,u-1(ϑ)(π)s,u⊤(x,p,ϑ)σs,u(p,ϑ)dWs(u)∫st‍Bs,u-1(ϑ)(π)s,u⊤(x,p,ϑ)×(bs,u(p,ϑ)+δs,u(p,ϑ)-rs,u(ϑ)1n)du
for all (x,p⊤,ϑ⊤)⊤∈𝕏(s) and 0≤s≤t≤T. In addition it is assumed that
(14)E[∫sT‍Hs,t(ϑ)cs,t(x,p,ϑ)dt]<∞,E[∫sT‍Hs,t(ϑ)Qs,t(p,ϑ)dt]<∞
for all (x,p⊤,ϑ⊤)⊤∈𝕏(s) and 0≤s≤t≤T. We say that ((π0,π),c,Q) as above is a portfolio evolution structure with rate of consumption cs,t and rate of endowment Q. We say (X,c,Q) as above is a hedgeable rate of consumption and endowment evolution structure with feasible set of values 𝕏. A (P,Θ)-consistent process Q (without any additional structure) that satisfies (14) is called a rate of endowment evolution process. If a wealth process (X,0,Q) is a hedgeable rate of consumption and endowment evolution structure with feasible set of values 𝕏, we just say that (X,Q) is a hedgeable endowment evolution structure.
A subsistence random fieldL for the market ℳ is a (P,Θ)-consistent process with drift and diffusion of class C0+, where Ls,·(p,ϑ)Hs,·(ϑ) is uniformly bounded below for all (p⊤,ϑ⊤)⊤∈𝔻(s) (where the bound might depend on p, ϑ, and s) such that
(15)E[Hs,t(ϑ)Ls,t(p,ϑ)]<∞
for all t. We will say that the couple (π,c) of portfolio on stocks and rate of consumption is admissible for (L,Q) and write (π,c)∈𝒜(L,Q) if for any (x,p⊤,ϑ⊤)⊤∈𝕏(s) with x≥Ls,s(p,ϑ)(16)Xs,t(x,p,ϑ)≥Ls,t(p,ϑ)∀t.
If there does not exist a couple of portfolio on stocks and rate of consumption admissible for (L,Q), we say that the class cited above is empty, and we would denote this by 𝒜(L,Q)=∅.


## 3. Consumption and Portfolio Optimization

Throughout this paper we are mainly interested in portfolio evolution structures that are obtained as the result of the optimal behavior of consumers as it is explained below. Next, we review definitions and properties of the type of utilities that are used in this paper.Definition 2.
Consider a functionU:(0,∞)↦ℝ is continuous, strictly increasing, strictly concave, and continuously differentiable with U′(∞)=limx→∞U′(x)=0 and U′(0+)≜limx↓0U′(x)=∞. Such a function will be called a utility function.Classic examples of utility functions areUα(x)=xα/α for some α∈(0,1), 0≤x<∞, and U(x)=log(x). For every utility function U(·), we will denote by I(·) the inverse of the derivative U′(·); both of these functions are continuous, strictly decreasing and map (0,∞) onto itself with I(0+)=U′(0+)=limx→0+U′(x)=∞, I(∞)=limx→∞I(x)=U′(∞)=0. We extend U by U(0)=U(0+) (we keep the same notation to the extension to [0,∞) of U hoping that it will be clear to the reader). It is a well-known result that
(17)max0<x<∞(U(x)-xy)=U(I(y))-yI(y),0<y<∞.Definition 3.
Consider a continuous functionU1:[0,T]×(0,∞)↦ℝ, such that U1(t,·) is a utility function in the sense of Definition 2 for all t∈[0,T]. It follows that I1(t,x)≜(∂U1(t,x)/∂x)-1, the inverse of the derivative of U, is a continuous function. Similarly, if a utility function U2:(0,∞)↦ℝ is given, then I2(x)≜(∂U2(x)/∂x)-1 is continuous. Let one denote
(18)𝒳(t,y)≜I2(y)+∫tT‍I1(t′,y)dt′.
We call a couple of functions U1 and U2 a state preference structure.Under the conditions outlined in the previous definition, it is easy to see that𝒳:[0,T]×(0,∞)→(0,∞) is a continuous function with the property that for each t, 𝒳(t,·) maps (0,∞) onto itself, strictly decreasing with 𝒳(t,0+)=limy↓0𝒳(t,y)=∞ and 𝒳(t,∞)=limy→∞𝒳(t,y)=0.We extendU1 and U2 by defining U1(t,0)=U(t,0+) for all 0≤t≤T and U2(0)=U2(0+), and we keep the same notation to the extension of U1 to [0,T]×[0,∞) and the extension of U2 to [0,∞).We point out that𝒳-1, defined for each t as 𝒳-1(t,·), the inverse of 𝒳(t,·), shares the same properties stated above about 𝒳. The discussion of those utility functions defined above, is given in Londoño [1].Fors≤t, define α(s,t)=𝒳(s,𝒳-1(t,·)). Then α(s,t)=α(s,t′)∘α(t′,t) for all s, t, and t′ in [0,T], where ∘ denotes standard composition of functions. We also observe that if αI(s,t)≜I1(s,𝒳-1(t,·)), then αI(s,t)∘α(t,s)=αI(s,s). Some examples discussed in Londoño [1] include power utility structures U1(t,x)=xαh(t) and U2(x)=cxα with α∈(0,1) and c≥0, where h:[0,T]→(0,∞) is any continuous positive function. Logarithmic utility structures are also included, where h is as above, U1(t,x)=h(t)log(x), and U2(x)=clog(x) with c≥0. Finally in the cited paper “separable preference structures” are introduced, where h is as above, U1(t,x)=h(t)u(x/h(t)), and U2(x)=cu(x/c), where u(·) is a utility function and c>0.Before we state the main result of this paper, we first require an additional definition.Definition 4.
Assume a rate of endowment evolution processQ of class C2,0+ with current value of future endowments L defined as
(19)Ls,t(p,ϑ)=-1Hs,t(ϑ)E[∫tT‍Hs,u(ϑ)Qs,u(p,ϑ)du∣ℱs,t]=-E[∫tT‍Ht,u(Θs,t(ϑ))Qt,u(Ps,t(p,ϑ),Θs,t(p,ϑ))du]=-Π(t,Ps,t(p,ϑ),Θs,t(ϑ)),
where it is assumed that
(20)Π(t,p,ϑ)≜-E[∫tT‍Ht,u(ϑ)Qt,u(p,ϑ)du]
is a function of class C2,0+. If (L,Q) is a hedgeable endowment evolution structure with portfolio πQ with feasible set of values 𝕏Q defined as
(21)𝕏Q={(s,x,p,ϑ)∣x=-Π(s,p,ϑ),(p⊤,ϑ⊤)⊤∈𝔻(s)},
one will say that Q is a hedgeable income structure.Remark 5.
We point out that unless the market is state complete (see Londoño, [15, Theorem 3.1] and Londoño [14]), in which case the volatility matrix has maximal range (in the sense that its rank is equal to n), there might be an infinity number of portfolios that hedge the given income structure. If the market is not necessarily complete but has no state-tame arbitrage opportunities, it satisfies (10) therefore by definition of wealth associated to a portfolio
(22)Bs,t-1(ϑ)Ls,t(p,ϑ)=x+∫st‍Bs,u-1(ϑ)Qs,u(p,ϑ)du+∫st‍Bs,u-1(ϑ)(σs,u⊤(p,ϑ)πs,u(x,p,ϑ))⊤dWs(u)+∫st‍Bs,u-1(ϑ)(σs,u⊤(p,ϑ)πs,u(x,p,ϑ))⊤θs,u(ϑ)du
for all (x,p⊤,ϑ⊤)⊤∈𝕏Q(s) and 0≤s≤t≤T, where Ls,t is the process in Definition 4. It follows that the same wealth is hedged by any portfolio πs,t+κs,t, where κs,t∈ker(σs,t⊤)≠∅.The following theorem extends the theory of optimal consumption and investment of Londoño [1] to incomplete markets which is the main result of this paper.Theorem 6.
Assume thatℳ=(P,Θ,𝔻,𝕂,b,σ,δ,θ,ρ,ϱ,r,p0,ϑ0) is a financial market that satisfies the smooth market condition (Condition 1) and (10). Also, assume that (U1,U2) is a state preference structure for a consumer with hedgeable income structure Q  (with hedging portfolio on the stocks πs,tQ).
Denote byαs,t and αtI the functions that correspond to the state preference structure (U1,U2) as defined after Definition 3. Define
(23)𝕏={(s,x,p⊤,ϑ⊤)⊤∣x>Π(s,p,ϑ),(p⊤,ϑ⊤)⊤∈𝔻(s)}.
Let ξ be defined as
(24)ξs,t(x,p,ϑ)≜Π(t,Ps,t(p,ϑ),Θs,t(ϑ))+Hs,t-1(ϑ)αt,s(x-Π(s,p,ϑ)),
and let c be defined as
(25)cs,t(x,p,ϑ)≜Hs,t-1(ϑ)(αtI∘αt,s)(x-Π(s,p,ϑ)
for any (x,p⊤,ϑ⊤)⊤∈𝕏(s). If ξ is a wealth evolution structure with feasible set of values 𝕏, then it is a hedgeable cumulative consumption and endowment structure with portfolio (π,c)∈𝒜(L,Q) which is an optimal solution for the problem of optimal consumption and investment. The optimality is in the sense that
(26)E[∫0T‍U1(t,H0,t(ϑ)c0,t(x,p,ϑ))dt+U2(H0,T(ϑ)ξ0,T(x,p,ϑ))∫0T‍]≥E[∫0T‍U1(t,H0,t(ϑ)c~0,t(x,p,ϑ))dt+U2(H0,T(ϑ)ξ~0,T(x,p,ϑ))∫0T‍]
for all (x,p⊤,ϑ⊤)⊤∈𝕏(s), where (ξ~,c~,Q) is any hedgeable cumulative consumption and endowment structure with (π~,c~)∈𝒜(L,Q), and
(27)E[∫0T‍U1-(t,H0,t(ϑ)c~0,t(x,p,ϑ))dt+U2-(H0,T(ϑ)ξ~0,T(x,p,ϑ))∫0T‍]<∞,
where U1-(t,x)=-(U1(t,x)∧0) and U2-(x)=-(U2(x)∧0). An optimal portfolio on stocks πs,t(x,p,ϑ) is
(28)αt,s(x-Π(s,p,ϑ))Hs,tκ(t,Ps,t,Θs,t)-πs,tQ(-Π(s,p,ϑ),p,ϑ),
where κ is the function defined by (11). Moreover, πs,t+ηs,t is an optimal portfolio for any (X,P,Θ)-consistent process ηs,t(x,p,ϑ)∈kerσs,t⊤(x,p,ϑ) for (x,p,ϑ)∈𝕏(s).Proof.
Define𝕏Q as the set given in Definition 4, and let πQ be the portfolio that hedges the income process. It follows that
(29)Hs,tLs,t-∫st‍Hs,uQs,udu=-E[∫sT‍Hs,uQs,udu∣ℱs,t]=-E[∫sT‍Hs,uQs,udu]+∫st‍Hs,u[σs,u⊤πQ-Ls,uθs,u]⊤dWs(u),
and on the other hand let us define πs,tC(x-Π(s,p,ϑ),p,ϑ) by the process that satisfies
(30)σs,t⊤πs,tC=αt,s(x-Π(s,p,ϑ))Hs,tθs,t.
The existence of this process follows by Condition 1. Therefore
(31)Hs,tHs,t-1αt,s(x-Π(s,p,ϑ))+∫st‍Hs,uHs,u-1αuI∘αu,s(x-Π(s,p,ϑ))du=αt,s(x-Π(s,p,ϑ))+∫st‍αuI∘αu,s(x-Π(s,p,ϑ))du=(x-Π(s,p,ϑ))=(x-Π(s,p,ϑ))+∫st‍Hs,u(σs,u⊤πs,uC-Hs,u-1αu,s(x-Π(s,p,ϑ))θs,u)⊤dWs(u).
Then
(32)Hs,tξs,t+∫st‍Hs,u(cs,u-Qs,u)du=x+∫st‍Hs,u(σs,u⊤(πs,uC-πs,uQ)-ξs,uθs.u)⊤dWs(u).
It follows as an application of Itô’s rule that πs,tC-πs,tQ is a portfolio that hedges ξs,t, and therefore (ξ,c,Q) is a hedgeable rate of consumption and endowment evolution structure. The proof that the consumption and endowment structure (ξ,c,Q) is optimal in the sense of the theorem is identical to the proof of [14, Theorem 2].Remark 7.
It is clear from the proof of Theorem6 that even if a sufficient number of stocks are added in order to make a complete market, then the consumption, wealth, and portfolio processes obtained in the above mentioned theorem are still optimal solutions.Remark 8.
We notice that the optimization problem described above is equivalent to the standard optimization problem for (state-dependent) utility random fields when the random field is defined as follows:(33)U1(t,c,ω)=U1(t,cH0,t(ω)),ω∈Ω,c>0,U2(x,ω)=U2(xH0,T(ω)),ω∈Ω,x>0,
where the definition of utility random fields is the one presented by Karatzas and Žitković [6, Definition 3.1]. However we notice that Karatzas and Žitković [6] require the random fields to be bounded, and this is not the case for our random fields; therefore their results do not include ours.As a result of the previous theorem, from a point of view of consumption of a consumer, he behaves as he sells the value of his income at the beginning and optimizes the consumption assuming no income at all.Corollary 9.
Assume the conditions and notation of Theorem6, and assume that the “augmented wealth”
(34)ξ′(x-Π(s,p,ϑ),p,ϑ)=ξs,t(x,p,ϑ)-Π(t,Ps,t(p,ϑ),Θs,t(ϑ)),
where ξs,t is the optimal wealth obtained in Theorem 6, is a wealth evolution structure with feasible set of values
(35)𝕐={(s,y,p⊤,ϑ⊤)⊤∣y>0,(p⊤,ϑ⊤)⊤∈𝔻(s)}.
Then, for any x>Π(s,p,ϑ) and (p⊤,ϑ⊤)⊤∈𝔻(s) and initial wealth y=x-Π(s,p,ϑ), the process of optimal hedgeable cumulative consumption and endowment structure (with no income) (ξ′,c,0) has the same optimal consumption as the one obtained in Theorem 6 (when it is assumed an income process Qs,t).Similar results to Theorem6 can be obtained from the problems of maximizing the expected utility from discounted consumption (alone) or from the problem of maximizing the expected utility from final wealth (alone) as was discussed in Londoño [14, Theorems 3 and 4]. The proof follows on the lines of Theorem 6.



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