State-Dependent Utilities and Incomplete Markets

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 includes incomplete 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.

1. Introduction

The problem of optimal consumption and investment for a “small investor” whose actions do not influence market prices in complete 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 obtained simple 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. Let 𝔻⊂[𝑠,𝑇]×ℝ^𝑘 be a measurable set for 0≤𝑠<𝑇, with section 𝔻(𝑡)={𝑥∈ℝ^𝑘∣(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→ℝ^𝑝 for some 𝑝<𝑘, where 𝑈⊂ℝ^𝑘 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 𝑋^𝑠₁,…,𝑋^𝑠_𝑘−𝑝∶[𝑠,𝑇]×𝑈→ℝ^𝑘, where 𝑈 is an open set as above, and 𝑋^𝑠₁(𝑡,⋅),…,𝑋^𝑠_𝑘−𝑝(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.

Let 𝑚 be a nonnegative integer, and let 𝔻 be a set defined as above. We say that 𝑓∈𝐶^𝑚,𝛿(𝔻∶ℝ^𝑛) if 𝑓 is a continuous function 𝑓∈𝐶(𝔻,ℝ^𝑛), and there exist an open set 𝕌 (relative to the topology of [0,𝑇]×ℝ^𝑘) with 𝔻⊂𝕌⊂[0,𝑇]×ℝ^𝑘 and a continuous extension 𝑓 of 𝑓 such that 𝑓(𝑡,⋅)∈𝐶^𝑚,𝛿(𝕌(𝑡)∶ℝ^𝑛), 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 ∫^𝑇_𝑠‖𝑓(𝑡,⋅)‖_{𝑚,𝛿∶𝐾}𝑑𝑡<∞, where 𝐾⊂𝕌 is a compact set, 0≤𝛿≤1. In case 𝑚=0 (or 𝛿=0), we denote 𝐶^𝑚,𝛿(𝔻∶ℝ^𝑛) simply by 𝐶^𝛿(𝔻∶ℝ^𝑛) (𝐶^𝑚(𝔻∶ℝ^𝑛)), and whenever clear we denote the above spaces simply as 𝐶^𝑚,𝛿, 𝐶^𝛿, and 𝐶^𝑚, respectively. We denote by 𝐶^𝑚,0+ the set ∪_𝛿>0𝐶^𝑚,𝛿, and 𝐶⁰⁺=𝐶^0,0+. Let 𝔻^′⊂[𝑠,𝑇]×ℝ^𝑘 be a measurable set with sections 𝔻^′(𝑡); we say that a function 𝑓 belongs to the class 𝐶^𝑚,𝛿(𝔻∶𝔻^′), where 𝛿>0 or 𝛿=0+, if 𝑓(𝑡,⋅) takes values in 𝔻^′(𝑡) and 𝑓∈𝐶^𝑚,𝛿(𝔻∶ℝ^𝑛). 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 a 𝑑-dimensional Brownian motion {𝑊(𝑡),ℱ_𝑡;0≤𝑡≤𝑇} starting at 0 and defined on a complete probability space (Ω,ℱ,𝐏), where ℱ=ℱ_𝑇 and {ℱ_𝑡,0≤𝑡≤𝑇} is the 𝐏 augmentation by the null sets of the natural filtration ℱ^𝑊_𝑡=𝜎(𝑊(𝑠),0≤𝑠≤𝑡). Let (ℱ_𝑠,𝑡)={ℱ_𝑠,𝑡,0≤𝑠≤𝑡≤𝑇} be the two-parameter filtration, where ℱ_𝑠,𝑡 is the smallest sub-𝜎-field containing all null sets and 𝜎(𝑊_𝑠(𝑢)∣𝑠≤𝑢≤𝑡), where 𝑊_𝑠(𝑢)≡𝑊(𝑢)−𝑊(𝑠). For each 0≤𝑠≤𝑇 we also define the 𝜎-field 𝒫_𝑠 of progressive measurable sets after time 𝑠 as the 𝜎-field of sets 𝑃∈ℬ([𝑠,𝑇])⊗ℱ_𝑠,𝑇, and the product 𝜎-field, such that 𝜒_𝑃(𝑡,𝜔), 𝑡≥𝑠, is a ℱ_𝑠,𝑡 progressive measurable (in 𝑡) process, where 𝜒 is the indicator function. We denote by 𝜇_𝑠 the measure on 𝒫_𝑠 defined by 𝜇_𝑠(𝑃)=𝐄∫^𝑇_𝑠𝜒_𝑃(𝑡,𝜔)𝑑𝑡.

Let 𝜑_𝑠,𝑡(𝑥,𝜔), 0≤𝑠≤𝑡≤𝑇, 𝑥∈𝔻(𝑠), be a ℝ^𝑛-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]×ℝ^𝑛 be a measurable set. We say that 𝜑_𝑠,𝑡 is a continuous 𝐶^𝑚,𝛿(𝔻∶𝔻^′)-semimartingale if 𝜑_𝑠∶𝑡→𝜑_𝑠,𝑡(⋅) is a continuous random field with values in 𝐶^𝑚,𝛿(𝔻∩([𝑠,𝑇]×ℝ^𝑘)∶𝔻^′∩([𝑠,𝑇]×ℝ^𝑛)), that is, a continuous (ℱ_𝑠,⋅) semimartingale process. In addition we assume that 𝜑_𝑠,𝑡(𝑥) can be decomposed as 𝜑_𝑠,𝑡(𝑥)=𝜑^loc_𝑠,𝑡(𝑥)+𝜑^fv_𝑠,𝑡(𝑥), where 𝜑^loc_𝑠,⋅(⋅) is a continuous 𝐶^𝑚,𝛿(𝔻∩([𝑠,𝑇]×ℝ^𝑘)∶𝔻^′∩([𝑠,𝑇]×ℝ^𝑛))-local-martingale, and 𝜑^fv_𝑠,⋅(⋅) is a continuous 𝐶^𝑚,𝛿(𝔻∩([𝑠,𝑇]×ℝ^𝑘)∶𝔻^′∩([𝑠,𝑇]×ℝ^𝑛))-process of bounded variation for each 0≤𝑠≤𝑇. A pair (𝑎,𝑏), where 𝑎_𝑠,𝑡(𝑥,𝑦) and 𝑏_𝑠,𝑡(𝑥) are measurable random fields ℱ_𝑠,𝑡-progressive measurable in 𝑡 for all 𝑥,𝑦∈𝔻(𝑠), 0≤𝑠≤𝑇, is said to be the local characteristics of 𝜑 if (𝑎_𝑠,⋅(𝑥,𝑦),𝑏_𝑠,⋅(𝑥)) is the local characteristic of 𝜑_𝑠≡𝜑_𝑠,⋅(⋅) for any 𝑠≤𝑇, (see Kunita [17]). In addition, a pair (𝜎,𝑏), where 𝜎_𝑠,𝑡(𝑥) is a measurable random field, that is, (ℱ_𝑠,𝑡)-progressive measurable in 𝑡 with values in the set 𝐿(ℝ^𝑑∶ℝ^𝑛) of real-valued matrices with size 𝑛×𝑑, and 𝑏 is as above, is said to be the diffusion and drift processes of 𝜑 if 𝜑^loc_𝑠,𝑡(𝑥)(𝜔)=^𝑡_𝑠𝜎_𝑠,𝑢(𝑥)𝑑𝑊_𝑠(𝑢) (1) for all 𝑥, 𝑠, 𝑡, and 𝜔. If 𝑏_𝑠,⋅(⋅) and 𝜎_𝑠,⋅(⋅) are processes of class 𝐶^𝑚,𝛿 for all 0≤𝑠≤𝑇, we will say that 𝜑 has diffusion and drift of class 𝐶^𝑚,𝛿.

Let 𝜑_𝑠,𝑡(𝑥) and 𝜓_𝑠,𝑡(𝑥) be continuous 𝐶(𝔻∶ℝ^𝑛) 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 𝑁_𝑠,𝑠′∈𝒫_𝑠′ with 𝜇_𝑠′(𝑁_𝑠,𝑠′)=0, such that 𝜑_𝑠,𝑡(𝑥)=𝜑_𝑠′,𝑡(𝜓_𝑠,𝑠′(𝑥)) for all (𝑡,𝜔)∉𝑁_𝑠,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 type 𝐶^𝑚,𝛿 on a manifold. Assume that 𝑏∶𝔻→ℝ^𝑛 and 𝜎∶𝔻→𝐿(ℝ^𝑛∶ℝ^𝑑) are functions in 𝐶^𝑚,𝛿, where 𝑚=0 and 𝛿=1 or 𝑚≥1 and 𝛿>0. Let 𝜑_𝑠,𝑡(𝑥) be the local (maximal) solution of 𝜑_𝑠,𝑡(𝑥)=𝑏̃𝑡,𝜑_𝑠,𝑡(𝑥+𝜎)𝑑𝑡𝑡,𝜑_𝑠,𝑡(𝑥)𝑑𝑊_𝑠(𝑡),𝜑_𝑠,𝑠(𝑥)=𝑥 (2) for 𝑥∈𝕌(𝑠), where 𝑏̃ and 𝜎 are the extensions of 𝑏 and 𝜎 to some open set 𝕌. We assume that 𝜑_𝑠,𝑡(𝑥) is the local solution of class 𝐶^𝑚,𝜖 for any 0≤𝜖<𝛿, see Kunita [17, Theorems 4.7.1 and 4.7.2]. If for any 𝑥∈𝔻(𝑠), 𝜑_𝑠,𝑡(𝑥) has a nonexplosive solution with values in 𝔻, we will say that 𝜑_𝑠,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 𝑏̃ and 𝜎 that are used.

Next, we describe a financial market. We assume an 𝑚-dimensional Itô process Θ of two parameters with values in 𝐶^2,0+(𝕂∶𝕂) (for 𝑚>1), where 𝕂(𝑠)⊂ℝ₊×ℝ^𝑚−1 (where ℝ₊=(0,∞)) for each 0≤𝑠≤𝑇, and 𝕂⊂[0,𝑇]×ℝ₊×ℝ^𝑚−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,…,𝜗_𝑚−1)^⊤∈𝕂(𝑠), 𝑑Θ⁰_𝑠,𝑡Θ⁰_𝑠,𝑡=𝑟𝑡,Θ_𝑠,𝑡𝑑𝑡+𝜃𝑡,Θ_𝑠,𝑡²𝑑𝑡+_{1≤𝑗≤𝑑}𝜃^𝑗𝑡,Θ_𝑠,𝑡𝑑𝑊^𝑗_𝑠(𝑡),Θ⁰_𝑠,𝑠(𝜗)=𝜗₀, (3) where ‖⋅‖ denotes the Euclidean norm, and Θ^𝑖_𝑠,𝑡=𝜌^𝑖𝑡,Θ_𝑠,𝑡𝑑𝑡+_{1≤𝑗≤𝑑}𝜚^𝑖,𝑗𝑡,Θ_𝑠,𝑡𝑑𝑊^𝑗_𝑠(𝑡)Θ^𝑖_𝑠,𝑠(𝜗)=𝜗_𝑖,𝑖=1,…,𝑚−1 (4) for some continuous functions 𝑟,𝜗^𝑗,𝜌^𝑖,𝜚^𝑖,𝑗 for 𝑗=1,…,𝑑 of class 𝐶^2,0+, and 𝑖=1,…,𝑚−1 for which global solutions of these stochastic differential equations exit. We also assume that for each (𝑝₁,…,𝑝_𝑛,𝜗₀,…,𝜗_𝑚−1)^⊤=(𝑝^⊤,𝜗^⊤)^⊤∈𝔻(𝑠), 𝑑𝑃^𝑖_𝑠,𝑡𝑃^𝑖_𝑠,𝑡=𝑏^𝑖𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡𝑑𝑡+_{1≤𝑗≤𝑑}𝜎^𝑖,𝑗𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡𝑑𝑊^𝑗_𝑠(𝑡)𝑃^𝑖_𝑠,𝑠𝑝,𝜗=𝑝_𝑖,𝑖=0,…,𝑛, (5) where, for 1≤𝑖≤𝑛, 𝑃^𝑖_𝑠,𝑡=𝜋^𝑖∘𝑃_𝑠,𝑡 is the price of shares outstanding for the 𝑖-stock and where 𝜋^𝑖 denotes the projection on the ith-component for 1≤𝑖≤𝑛. We assume that 𝜎^𝑖,𝑗, 𝑏^𝑖 are continuous functions of class 𝐶^2,0+ 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/Θ⁰_𝑠,𝑡 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 𝐻_𝑠,𝑡 below.

The process of bounded variation 𝐵_𝑠,𝑡≡𝐵_𝑠,𝑡(𝜗), whose evolution is given by the stochastic differential equation 𝑑𝐵_𝑠,𝑡=𝑟𝑡,Θ_𝑠,𝑡𝐵_𝑠,𝑡𝑑𝑡,𝐵_𝑠,𝑠=1,for𝑠≤𝑡≤𝑇, (6) will be called the bond price process.

We define the state price density process 𝐻_𝑠,𝑡≡𝐻_𝑠,𝑡(𝜗) to be the continuous 𝐶⁰⁺(𝕂∶ℝ₊) process given by 𝐻_𝑠,𝑡=𝐵⁻¹_𝑠,𝑡𝑍_𝑠,𝑡,𝑠≤𝑡≤𝑇, (7) where 𝑍_𝑠,𝑡≡𝑍_𝑠,𝑡(𝜗) is the process defined as 𝑍_𝑠,𝑡=exp−^𝑡_𝑠𝜃^⊤𝑢,Θ_𝑠,𝑢(𝑢)−1𝑑𝑊2^𝑡_𝑠𝜃𝑢,Θ_𝑠,𝑢²𝑑𝑢 (8) for 𝑠≤𝑡≤𝑇, and 𝐵⁻¹_𝑠,𝑡=1/𝐵_𝑠,𝑡. We point out that 𝐻_𝑠,𝑡=𝜗₀Θ⁰_𝑠,𝑡 (9) 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 𝑏𝑡,𝑝,𝜗+𝛿𝑡,𝑝,𝜗−𝑟(𝑡,𝜗)𝟏_𝑛=𝜎𝑡,𝑝,𝜗𝜃(𝑡,𝜗) (10) for all (𝑝^⊤,𝜗^⊤)^⊤∈𝔻(𝑠) and 𝑡, where 𝟏^⊤_𝑛=(1,…,1)∈ℝ^𝑛. 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 a 𝐶^2,0+-matrix-valued function 𝜅 defined on 𝔻 with the property that 𝜎^⊤𝑡,𝑝,𝜃𝜅𝑡,𝑝,𝜗=𝜃(𝑡,𝜗). (11)
We point out that we require no dependence of prices on the evolution of the functions 𝑟 and 𝜃. We say that 𝑏_𝑠,𝑡≡𝑏(𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡) is the return process, 𝜎_𝑠,𝑡≡𝜎(𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡) is the volatility process, 𝛿_𝑠,𝑡≡𝛿(𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡) is the process of dividends, 𝑟_𝑠,𝑡≡𝑟(𝑡,Θ_𝑠,𝑡) is the interest rate process, and 𝜃_𝑠,𝑡≡𝜃(𝑡,Θ_𝑠,𝑡) is the market price of risk process.
For a structure as above, we say that ℳ=(𝑃,Θ,𝔻,𝕂,𝑏,𝜎,𝛿,𝜃,𝜌,𝜚,𝑟,𝑝⁰,𝜗⁰) is a financial market with terminal time 𝑇 and initial time 0, feasible set of values 𝔻, feasible set of state values 𝕂, vector of returns 𝑏(𝑡,⋅)=(𝑏₁(𝑡,⋅)…,𝑏_𝑛(𝑡,⋅))^⊤, matrix of volatility coefficients 𝜎(𝑡,⋅)=(𝜎^𝑖,𝑗(𝑡,⋅)), vector of dividends 𝛿(𝑡,⋅)=(𝛿₁(𝑡,⋅),…,𝛿_𝑛(𝑡,⋅))^⊤, market price of risk 𝜃(𝑡,⋅)=(𝜃₁(𝑡,⋅),…,𝜃_𝑑(𝑡,⋅))^⊤, interest rate 𝑟(𝑡,⋅), drift of the state process 𝜌(𝑡,⋅)=(𝜌¹(𝑡,⋅),…,𝜌^𝑚(𝑡,⋅))^⊤, diffusion matrix for the state process 𝜚(𝑡,⋅)=(𝜚^𝑖,𝑗(𝑡,⋅)), vector of initial prices 𝑝⁰∈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 set 𝕏⊂[0,𝑇]×ℝ^𝑛+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 𝑋=𝑋_𝑠,𝑡𝑥,𝑝,𝜗;𝑥,𝑝^⊤,𝜗^⊤⊤∈𝕏(𝑠),0≤𝑠≤𝑡≤𝑇, (12) 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)={(𝜋⁰_𝑠,𝑡(𝑥,𝑝,𝜗),…,𝜋^𝑛_𝑠,𝑡(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)^⊤𝟏_𝑛=𝑋. Assume that 𝑐={𝑐_𝑠,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={𝑄_𝑠,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 𝐵⁻¹_𝑠,𝑡(𝜗)𝑋_𝑠,𝑡𝑥,𝑝,𝜗=𝑥+^𝑡_𝑠𝐵⁻¹_𝑠,𝑢(𝜗)𝑄_𝑠,𝑢𝑝,𝜗−𝑐_𝑠,𝑢+𝑥,𝑝,𝜗𝑑𝑢_𝑡𝑠𝐵⁻¹_𝑠,𝑢(𝜗)(𝜋)^⊤_𝑠,𝑢𝑥,𝑝,𝜗𝜎_𝑠,𝑢𝑝,𝜗𝑑𝑊_𝑠(𝑢)_𝑡𝑠𝐵⁻¹_𝑠,𝑢(𝜗)(𝜋)^⊤_𝑠,𝑢×𝑏𝑥,𝑝,𝜗_𝑠,𝑢𝑝,𝜗+𝛿_𝑠,𝑢𝑝,𝜗−𝑟_𝑠,𝑢(𝜗)𝟏_𝑛𝑑𝑢 (13) for all (𝑥,𝑝^⊤,𝜗^⊤)^⊤∈𝕏(𝑠) and 0≤𝑠≤𝑡≤𝑇. In addition it is assumed that 𝐄^𝑇_𝑠𝐻_𝑠,𝑡(𝜗)𝑐_𝑠,𝑡𝐄𝑥,𝑝,𝜗𝑑𝑡<∞,_𝑇𝑠𝐻_𝑠,𝑡(𝜗)𝑄_𝑠,𝑡𝑝,𝜗𝑑𝑡<∞ (14) 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 𝑐_𝑠,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 field 𝐿 for the market ℳ is a (𝑃,Θ)-consistent process with drift and diffusion of class 𝐶⁰⁺, where 𝐿_𝑠,⋅(𝑝,𝜗)𝐻_𝑠,⋅(𝜗) is uniformly bounded below for all (𝑝^⊤,𝜗^⊤)^⊤∈𝔻(𝑠) (where the bound might depend on 𝑝, 𝜗, and 𝑠) such that 𝐄𝐻_𝑠,𝑡(𝜗)𝐿_𝑠,𝑡𝑝,𝜗<∞ (15) 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≥𝐿_𝑠,𝑠(𝑝,𝜗)𝑋_𝑠,𝑡𝑥,𝑝,𝜗≥𝐿_𝑠,𝑡𝑝,𝜗∀𝑡. (16) 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 function 𝑈∶(0,∞)↦ℝ is continuous, strictly increasing, strictly concave, and continuously differentiable with 𝑈^′(∞)=lim_𝑥→∞𝑈^′(𝑥)=0 and 𝑈^′(0+)≜lim_𝑥↓0𝑈^′(𝑥)=∞. Such a function will be called a utility function.

Classic examples of utility functions are 𝑈_𝛼(𝑥)=𝑥^𝛼/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+)=lim_𝑥→0⁺𝑈^′(𝑥)=∞, 𝐼(∞)=lim_𝑥→∞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 max_0<𝑥<∞𝑈(𝑥)−𝑥𝑦=𝑈𝐼𝑦−𝑦𝐼𝑦,0<𝑦<∞. (17)

Definition 3. Consider a continuous function 𝑈₁∶[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 𝒳𝑡,𝑦≜𝐼₂𝑦+^𝑇_𝑡𝐼₁𝑡^′,𝑦𝑑𝑡^′. (18) 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 that 𝒳∶[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+)=lim_𝑦↓0𝒳(𝑡,𝑦)=∞ and 𝒳(𝑡,∞)=lim_𝑦→∞𝒳(𝑡,𝑦)=0.

We extend 𝑈₁ 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 that 𝒳⁻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].

For 𝑠≤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 𝛼^𝐼(𝑠,𝑡)≜𝐼₁(𝑠,𝒳⁻¹(𝑡,⋅)), then 𝛼^𝐼(𝑠,𝑡)∘𝛼(𝑡,𝑠)=𝛼^𝐼(𝑠,𝑠). Some examples discussed in Londoño [1] include power utility structures 𝑈₁(𝑡,𝑥)=𝑥^𝛼ℎ(𝑡) and 𝑈₂(𝑥)=𝑐𝑥^𝛼 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 process 𝑄 of class 𝐶^2,0+ with current value of future endowments 𝐿 defined as 𝐿_𝑠,𝑡=𝑝,𝜗−1𝐻_𝑠,𝑡(𝜗)𝐄^𝑇_𝑡𝐻_𝑠,𝑢(𝜗)𝑄_𝑠,𝑢𝑝,𝜗𝑑𝑢∣ℱ_𝑠,𝑡=−𝐄^𝑇_𝑡𝐻_𝑡,𝑢Θ_𝑠,𝑡(𝜗)𝑄_𝑡,𝑢𝑃_𝑠,𝑡𝑝,𝜗,Θ_𝑠,𝑡𝑝,𝜗𝑑𝑢=−Π𝑡,𝑃_𝑠,𝑡𝑝,𝜗,Θ_𝑠,𝑡(𝜗), (19) where it is assumed that Π𝑡,𝑝,𝜗≜−𝐄^𝑇_𝑡𝐻_𝑡,𝑢(𝜗)𝑄_𝑡,𝑢𝑝,𝜗𝑑𝑢 (20) is a function of class 𝐶^2,0+. If (𝐿,𝑄) is a hedgeable endowment evolution structure with portfolio 𝜋_𝑄 with feasible set of values 𝕏_𝑄 defined as 𝕏_𝑄=𝑠,𝑥,𝑝,𝜗∣𝑥=−Π𝑠,𝑝,𝜗,𝑝^⊤,𝜗^⊤⊤∈𝔻(𝑠), (21) 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 𝐵⁻¹_𝑠,𝑡(𝜗)𝐿_𝑠,𝑡𝑝,𝜗=𝑥+^𝑡_𝑠𝐵⁻¹_𝑠,𝑢(𝜗)𝑄_𝑠,𝑢+𝑝,𝜗𝑑𝑢_𝑡𝑠𝐵⁻¹_𝑠,𝑢(𝜗)𝜎^⊤_𝑠,𝑢𝑝,𝜗𝜋_𝑠,𝑢𝑥,𝑝,𝜗^⊤𝑑𝑊_𝑠(𝑢)+^𝑡_𝑠𝐵⁻¹_𝑠,𝑢(𝜗)𝜎^⊤_𝑠,𝑢𝑝,𝜗𝜋_𝑠,𝑢𝑥,𝑝,𝜗^⊤𝜃_𝑠,𝑢(𝜗)𝑑𝑢 (22) for all (𝑥,𝑝^⊤,𝜗^⊤)^⊤∈𝕏_𝑄(𝑠) and 0≤𝑠≤𝑡≤𝑇, where 𝐿_𝑠,𝑡 is the process in Definition 4. It follows that the same wealth is hedged by any portfolio 𝜋_𝑠,𝑡+𝜅_𝑠,𝑡, where 𝜅_𝑠,𝑡∈ker(𝜎^⊤_𝑠,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 that ℳ=(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 𝜋^𝑄_𝑠,𝑡).
Denote by 𝛼_𝑠,𝑡 and 𝛼^𝐼_𝑡 the functions that correspond to the state preference structure (𝑈₁,𝑈₂) as defined after Definition 3. Define 𝕏=𝑠,𝑥,𝑝^⊤,𝜗^⊤⊤∣𝑥>Π𝑠,𝑝,𝜗,𝑝^⊤,𝜗^⊤⊤∈𝔻(𝑠). (23) Let 𝜉 be defined as 𝜉_𝑠,𝑡𝑥,𝑝,𝜗≜Π𝑡,𝑃_𝑠,𝑡𝑝,𝜗,Θ_𝑠,𝑡(𝜗)+𝐻⁻¹_𝑠,𝑡(𝜗)𝛼_𝑡,𝑠𝑥−Π𝑠,𝑝,𝜗, (24) and let 𝑐 be defined as 𝑐_𝑠,𝑡𝑥,𝑝,𝜗≜𝐻⁻¹_𝑠,𝑡(𝜗)𝛼^𝐼_𝑡∘𝛼_𝑡,𝑠𝑥−Π𝑠,𝑝,𝜗 (25) 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 𝐄^𝑇₀𝑈₁𝑡,𝐻_0,𝑡(𝜗)𝑐_0,𝑡𝑥,𝑝,𝜗𝑑𝑡+𝑈₂𝐻_0,𝑇(𝜗)𝜉_0,𝑇𝑥,𝑝,𝜗≥𝐄^𝑇₀𝑈₁𝑡,𝐻_0,𝑡(𝜗)𝑐̃_0,𝑡𝑥,𝑝,𝜗𝑑𝑡+𝑈₂𝐻_0,𝑇(𝜗)𝜉̃_0,𝑇𝑥,𝑝,𝜗 (26) for all (𝑥,𝑝^⊤,𝜗^⊤)^⊤∈𝕏(𝑠), where (𝜉̃,𝑐,̃𝑄) is any hedgeable cumulative consumption and endowment structure with (𝜋,𝑐)̃∈𝒜(𝐿,𝑄), and 𝐄^𝑇₀𝑈⁻₁𝑡,𝐻_0,𝑡(𝜗)𝑐̃_0,𝑡𝑥,𝑝,𝜗𝑑𝑡+𝑈⁻₂𝐻_0,𝑇(𝜗)𝜉̃_0,𝑇𝑥,𝑝,𝜗<∞, (27) where 𝑈⁻₁(𝑡,𝑥)=−(𝑈₁(𝑡,𝑥)∧0) and 𝑈⁻₂(𝑥)=−(𝑈₂(𝑥)∧0). An optimal portfolio on stocks 𝜋_𝑠,𝑡(𝑥,𝑝,𝜗) is 𝛼_𝑡,𝑠𝑥−Π𝑠,𝑝,𝜗𝐻_𝑠,𝑡𝜅𝑡,𝑃_𝑠,𝑡,Θ_𝑠,𝑡−𝜋^𝑄_𝑠,𝑡−Π𝑠,𝑝,𝜗,𝑝,𝜗, (28) where 𝜅 is the function defined by (11). Moreover, 𝜋_𝑠,𝑡+𝜂_𝑠,𝑡 is an optimal portfolio for any (𝑋,𝑃,Θ)-consistent process 𝜂_𝑠,𝑡(𝑥,𝑝,𝜗)∈ker𝜎^⊤_𝑠,𝑡(𝑥,𝑝,𝜗) for (𝑥,𝑝,𝜗)∈𝕏(𝑠).

Proof. Define 𝕏_𝑄 as the set given in Definition 4, and let 𝜋_𝑄 be the portfolio that hedges the income process. It follows that 𝐻_𝑠,𝑡𝐿_𝑠,𝑡−^𝑡_𝑠𝐻_𝑠,𝑢𝑄_𝑠,𝑢𝑑𝑢=−𝐄^𝑇_𝑠𝐻_𝑠,𝑢𝑄_𝑠,𝑢𝑑𝑢∣ℱ_𝑠,𝑡=−𝐄^𝑇_𝑠𝐻_𝑠,𝑢𝑄_𝑠,𝑢+𝑑𝑢_𝑡𝑠𝐻_𝑠,𝑢𝜎^⊤_𝑠,𝑢𝜋_𝑄−𝐿_𝑠,𝑢𝜃_𝑠,𝑢^⊤𝑑𝑊_𝑠(𝑢), (29) and on the other hand let us define 𝜋^𝐶_𝑠,𝑡(𝑥−Π(𝑠,𝑝,𝜗),𝑝,𝜗) by the process that satisfies 𝜎^⊤_𝑠,𝑡𝜋^𝐶_𝑠,𝑡=𝛼_𝑡,𝑠𝑥−Π𝑠,𝑝,𝜗𝐻_𝑠,𝑡𝜃_𝑠,𝑡. (30) The existence of this process follows by Condition 1. Therefore 𝐻_𝑠,𝑡𝐻⁻¹_𝑠,𝑡𝛼_𝑡,𝑠+𝑥−Π𝑠,𝑝,𝜗_𝑡𝑠𝐻_𝑠,𝑢𝐻⁻¹_𝑠,𝑢𝛼^𝐼_𝑢∘𝛼_𝑢,𝑠𝑥−Π𝑠,𝑝,𝜗𝑑𝑢=𝛼_𝑡,𝑠+𝑥−Π𝑠,𝑝,𝜗_𝑡𝑠𝛼^𝐼_𝑢∘𝛼_𝑢,𝑠==+𝑥−Π𝑠,𝑝,𝜗𝑑𝑢𝑥−Π𝑠,𝑝,𝜗𝑥−Π𝑠,𝑝,𝜗_𝑡𝑠𝐻_𝑠,𝑢𝜎^⊤_𝑠,𝑢𝜋^𝐶_𝑠,𝑢−𝐻⁻¹_𝑠,𝑢𝛼_𝑢,𝑠𝑥−Π𝑠,𝑝,𝜗𝜃_𝑠,𝑢^⊤𝑑𝑊_𝑠(𝑢). (31) Then 𝐻_𝑠,𝑡𝜉_𝑠,𝑡+^𝑡_𝑠𝐻_𝑠,𝑢𝑐_𝑠,𝑢−𝑄_𝑠,𝑢𝑑𝑢=𝑥+^𝑡_𝑠𝐻_𝑠,𝑢𝜎^⊤_𝑠,𝑢(𝜋^𝐶_𝑠,𝑢−𝜋^𝑄_𝑠,𝑢)−𝜉_𝑠,𝑢𝜃_𝑠.𝑢^⊤𝑑𝑊_𝑠(𝑢). (32) It follows as an application of Itô’s rule that 𝜋^𝐶_𝑠,𝑡−𝜋^𝑄_𝑠,𝑡 is a portfolio that hedges 𝜉_𝑠,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 Theorem 6 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: 𝑈₁(𝑡,𝑐,𝜔)=𝑈₁𝑡,𝑐𝐻_0,𝑡(𝜔)𝑈,𝜔∈Ω,𝑐>0,₂(𝑥,𝜔)=𝑈₂𝑥𝐻_0,𝑇(𝜔),𝜔∈Ω,𝑥>0, (33) 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 Theorem 6, and assume that the “augmented wealth” 𝜉^′𝑥−Π𝑠,𝑝,𝜗,𝑝,𝜗=𝜉_𝑠,𝑡𝑥,𝑝,𝜗−Π𝑡,𝑃_𝑠,𝑡𝑝,𝜗,Θ_𝑠,𝑡(𝜗), (34) where 𝜉_𝑠,𝑡 is the optimal wealth obtained in Theorem 6, is a wealth evolution structure with feasible set of values 𝕐=𝑠,𝑦,𝑝^⊤,𝜗^⊤⊤∣𝑦>0,𝑝^⊤,𝜗^⊤⊤∈𝔻(𝑠). (35) 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 𝑄_𝑠,𝑡).

Similar results to Theorem 6 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.

Acknowledgment

The authors would like to thank Professor Darrell Duffie for the detailed reading of this paper, positive and encouraging comments, and valuable suggestions that led to a substantial improvement of the paper. All remaining errors are the author’s.