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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 359701, 8 pages
State-Dependent Utilities and Incomplete Markets
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Received 23 January 2013; Accepted 1 May 2013
Academic Editor: Guangchen Wang
Copyright © 2013 Jaime A. Londoño. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 . 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  and Karatzas et al.  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  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 , Hugonnier and Kramkov , and Cvitanić et al. . 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  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 . This result is even true for semimartingale complete markets as is pointed out in Karatzas and Žitković .
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 , Breeden , Cox et al. , or Londoño  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 . 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 . 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  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 . In the spirit of Merton , Breeden , and Cox et al.  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 .
This paper also depends on the valuation and arbitrage theory presented in Londoño ; this is an extension of the theory of state tameness (Londoño ) 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  and the references therein.
2. The Model
First we introduce some notation which will be frequently used in this paper. Let be a measurable set for , with section for . We assume that for each , has a differential structure, which at this point is not necessary to specify. Examples of this differential structure are sets whose sections are the solution sets of a function for some , where is an open set, and 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 has maximal rank for each . Other examples are sets whose sections are integral manifolds defined by () continuous vector fields , where is an open set as above, and 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 . 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 be a nonnegative integer, and let be a set defined as above. We say that if is a continuous function , and there exist an open set (relative to the topology of ) with and a continuous extension of such that , where this last space is the Fréchet space of -times continuous differentiable functions whose mth-order derivatives are -Hölder continuous with seminorms defined in Kunita [10, Section 3.1] with , where is a compact set, . In case (or ), we denote simply by (), and whenever clear we denote the above spaces simply as , , and , respectively. We denote by the set , and . Let be a measurable set with sections ; we say that a function belongs to the class , where or , if takes values in and . 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 .
We assume a -dimensional Brownian motion starting at and defined on a complete probability space , where and is the augmentation by the null sets of the natural filtration . Let be the two-parameter filtration, where is the smallest sub--field containing all null sets and , where . For each we also define the -field of progressive measurable sets after time as the -field of sets , and the product -field, such that , , is a progressive measurable (in ) process, where is the indicator function. We denote by the measure on defined by .
Let , , , be a -valued random field on the probability space , where has a differential structure sufficiently smooth for the following definitions to make sense, and let be a measurable set. We say that is a continuous -semimartingale if is a continuous random field with values in , that is, a continuous semimartingale process. In addition we assume that can be decomposed as , where is a continuous -local-martingale, and is a continuous -process of bounded variation for each . A pair , where and are measurable random fields -progressive measurable in for all , , is said to be the local characteristics of if is the local characteristic of for any , (see Kunita ). In addition, a pair , where is a measurable random field, that is, -progressive measurable in with values in the set of real-valued matrices with size , and is as above, is said to be the diffusion and drift processes of if for all , , , and . If and are processes of class for all , we will say that has diffusion and drift of class .
Let and be continuous and semimartingales, respectively, where in addition, it is assumed that for all . We say that the process is a -consistent semi-martingale process if for each there exists a set with , such that for all and all . We say that the process is a consistent semi-martingale process if is a -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 on a manifold. Assume that and are functions in , where and or and . Let be the local (maximal) solution of for , where and are the extensions of and to some open set . We assume that is the local solution of class for any , see Kunita [17, Theorems 4.7.1 and 4.7.2]. If for any , has a nonexplosive solution with values in , we will say that , is the solution to the stochastic differential equation on . It is straightforward to see that this defines uniquely a process that does not depend on the extensions and that are used.
Next, we describe a financial market. We assume an -dimensional Itô process of two parameters with values in (for ), where (where ) for each , and is some measurable set with a differential structure as discussed above. We assume an -dimensional Itô process of two parameters with values in , where is a measurable set with sections for each , where and with a differential structure as discussed above. We assume that for each initial condition , where denotes the Euclidean norm, and for some continuous functions for of class , and for which global solutions of these stochastic differential equations exit. We also assume that for each , where, for , is the price of shares outstanding for the -stock and where denotes the projection on the ith-component for . We assume that , are continuous functions of class 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, is the process (with two parameters) that discounts the flow of money in every future state , to bring the value of the flow to current time ; for instance see Londoño  and the definition of below.
The process of bounded variation , whose evolution is given by the stochastic differential equation will be called the bond price process.
We define the state price density process to be the continuous process given by where is the process defined as for , and . We point out that easily follows the Itô’s lemma. Throughout this paper we will assume that , where denotes the orthogonal complement of the kernel of and for all and , where . This latter assumption implies that there are no state-tame arbitrage opportunities (see Londoño ). 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 , the existence of a measurable function 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 -matrix-valued function defined on with the property that
We point out that we require no dependence of prices on the evolution of the functions and . We say that is the return process, is the volatility process, is the process of dividends, is the interest rate process, and is the market price of risk process.
For a structure as above, we say that is a financial market with terminal time and initial time 0, feasible set of values , feasible set of state values , vector of returns , matrix of volatility coefficients , vector of dividends , market price of risk , interest rate , drift of the state process , diffusion matrix for the state process , vector of initial prices , and initial state variables .
Next, we review and extend some definitions from Londoño  that are needed to describe equilibrium.
Definition 1. Assume a measurable set with some differential structure as discussed in Section 2 with (nonempty) section for each . Assume a family of continuous Itô’s processes
such that is a consistent process of class . We say that is a wealth evolution structure; we will denote this by writing . For a detailed description of consistent processes and related ones see Londoño [1, 14]. One says that is a feasible set of values for .
Let ; be a -consistent progressive measurable process of class with . Assume that is a nonnegative consistent progressive measurable process of class and that is a non-negative consistent progressive measurable process of class . It is assumed that for all and . In addition it is assumed that for all and . We say that as above is a portfolio evolution structure with rate of consumption and rate of endowment . We say as above is a hedgeable rate of consumption and endowment evolution structure with feasible set of values . A -consistent process (without any additional structure) that satisfies (14) is called a rate of endowment evolution process. If a wealth process is a hedgeable rate of consumption and endowment evolution structure with feasible set of values , we just say that is a hedgeable endowment evolution structure.
A subsistence random field for the market is a -consistent process with drift and diffusion of class , where is uniformly bounded below for all (where the bound might depend on , , and ) such that for all . We will say that the couple of portfolio on stocks and rate of consumption is admissible for and write if for any with If there does not exist a couple of portfolio on stocks and rate of consumption admissible for , we say that the class cited above is empty, and we would denote this by .
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 is continuous, strictly increasing, strictly concave, and continuously differentiable with and . Such a function will be called a utility function.
Classic examples of utility functions are for some , , and . For every utility function , we will denote by the inverse of the derivative ; both of these functions are continuous, strictly decreasing and map onto itself with , . We extend by (we keep the same notation to the extension to of hoping that it will be clear to the reader). It is a well-known result that
Definition 3. Consider a continuous function , such that is a utility function in the sense of Definition 2 for all . It follows that , the inverse of the derivative of , is a continuous function. Similarly, if a utility function is given, then is continuous. Let one denote We call a couple of functions and a state preference structure.
Under the conditions outlined in the previous definition, it is easy to see that is a continuous function with the property that for each , maps onto itself, strictly decreasing with and .
We extend and by defining for all and , and we keep the same notation to the extension of to and the extension of to .
We point out that , defined for each as , the inverse of , shares the same properties stated above about . The discussion of those utility functions defined above, is given in Londoño .
For , define . Then for all , , and in , where denotes standard composition of functions. We also observe that if , then . Some examples discussed in Londoño  include power utility structures and with and , where is any continuous positive function. Logarithmic utility structures are also included, where is as above, , and with . Finally in the cited paper “separable preference structures” are introduced, where is as above, , and , where is a utility function and .
Before we state the main result of this paper, we first require an additional definition.
Definition 4. Assume a rate of endowment evolution process of class with current value of future endowments defined as where it is assumed that is a function of class . If is a hedgeable endowment evolution structure with portfolio with feasible set of values defined as one will say that 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 ), in which case the volatility matrix has maximal range (in the sense that its rank is equal to ), 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 for all and , where is the process in Definition 4. It follows that the same wealth is hedged by any portfolio , where .
The following theorem extends the theory of optimal consumption and investment of Londoño  to incomplete markets which is the main result of this paper.
Theorem 6. Assume that is a financial market that satisfies the smooth market condition (Condition 1) and (10). Also, assume that is a state preference structure for a consumer with hedgeable income structure with hedging portfolio on the stocks .
Denote by and the functions that correspond to the state preference structure as defined after Definition 3. Define Let be defined as and let be defined as for any . If is a wealth evolution structure with feasible set of values , then it is a hedgeable cumulative consumption and endowment structure with portfolio which is an optimal solution for the problem of optimal consumption and investment. The optimality is in the sense that for all , where is any hedgeable cumulative consumption and endowment structure with , and where and . An optimal portfolio on stocks is where is the function defined by (11). Moreover, is an optimal portfolio for any -consistent process for .
Proof. Define as the set given in Definition 4, and let be the portfolio that hedges the income process. It follows that and on the other hand let us define by the process that satisfies The existence of this process follows by Condition 1. Therefore Then It follows as an application of Itô’s rule that is a portfolio that hedges , and therefore is a hedgeable rate of consumption and endowment evolution structure. The proof that the consumption and endowment structure 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: 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  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” where is the optimal wealth obtained in Theorem 6, is a wealth evolution structure with feasible set of values Then, for any and and initial wealth , the process of optimal hedgeable cumulative consumption and endowment structure (with no income) has the same optimal consumption as the one obtained in Theorem 6 (when it is assumed an income process ).
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.
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.
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