# Utility maximization with addictive consumption habit formation in incomplete semimartingale markets\thanksrefT1

###### Abstract

This paper studies the continuous time utility maximization problem on consumption with addictive habit formation in incomplete semimartingale markets. Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into a time-separable utility maximization problem with a shadow random endowment on the product space . Existence and uniqueness of the optimal solution are established using convex duality approach, where the primal value function is defined on two variables, that is, the initial wealth and the initial standard of living. We also provide sufficient conditions on the stochastic discounting processes and on the utility function for the well-posedness of the original optimization problem. Under the same assumptions, classical proofs in the approach of convex duality analysis can be modified when the auxiliary dual process is not necessarily integrable.

10.1214/14-AAP1026 \volume25 \issue3 2015 \firstpage1383 \lastpage1419 \newproclaimdefinitionDefinition[section] \newproclaimassumptionAssumption[section] \newproclaimremarkRemark

Habit formation

A]\fnmsXiang \snmYu\correflabel=e1]xymath@umich.edu \thankstextT1This work is part of the author’s Ph.D. dissertation at the University of Texas at Austin and is partially supported by NSF Grant under the award number DMS-09-08441.

class=AMS] \kwd[Primary ]91G10 \kwd91B42 \kwd[; secondary ]91G80 Time nonseparable utility maximization \kwdconsumption habit formation \kwdauxiliary processes \kwdconvex duality \kwdincomplete markets

## 1 Introduction

During the past decades, the time separable von Neumann–Morgenstern preferences on consumption have been observed to be inconsistent with some empirical evidences. For instance, the well-known magnitude of the equity premium (Mehra and Prescott Mehra ) cannot be reconciled with the preference where the instantaneous utility function is only derived from the consumption rate process. As an alternative modeling tool, linear addictive habit formation preference has attracted a lot of attention and has been actively investigated in recent years. This new preference is defined by , where and the additional accumulative process , called the habit formation or the standard of living process, describes the consumption history impact. To be more precise, is the solution of the following recursive equation:

where the discounting factors and are assumed to be nonnegative optional processes, and the given real number is called the initial habit or the initial standard of living. Moreover, the consumption habits are assumed to be addictive in the sense that for all time . Compared to the time separable case, a small drop in consumption may cause large fluctuation in consumption net of the subsistence level due to the standard of living constraint. The habit formation preference can possibly explain sizable excess returns on risky assets in equilibrium models even for moderate values of the degree of risk aversion. Based on this, a vast literature recommends this time nonseparable preference as the new economic paradigm. We refer readers to, for instance, Constantinides constantinides1988habit , Samuelson RePEccc and Campbell and Cochrane Campbell .

The study of habit formation in modern economics dates back to Hicks Hicks in and Ryder and Heal Heal in . More recently, there have been some important contributions in complete Itô processes markets; see Detemple and Zapatero detemple91 , detemple92 , Schroder and Skiadas Schroder01072002 , Munk Munk , Detemple and Karatzas Detemple2003265 and Englezos and Karatzas Eng09 . Several pioneering work have derived the explicit feedback form of the optimal policies under different assumptions and market models. However, in the words by Englezos and Karatzas Eng09 , “The existence of an optimal portfolio/consumption pair in an incomplete market is an open question …, and new methodologies are needed to handle the problem.” Therefore, in this paper, we are interested in the general incomplete semimartingale framework and aim to prove the existence and uniqueness of the optimal solution to this path dependent optimization problem.

The convex duality approach plays an important role in the treatment of utility maximization problems in incomplete markets. To list a very small subset of the existing literature, we refer to Karatzas et al. KKK , Kramkov and Schachermayer kram99 , kram03 , Cvitanic, Schachermayer and Wang Cvitanic , Karatzas and Žitković KarZit03 , Hugonnier and Kramkov kram04 and Žitković Zit02 , Zit05 .

Typically, the critical step to build conjugate duality is to define the dual space as a proper extension of space , which is the set of equivalent local martingale measure density processes. The first natural choice is the bipolar set of the space , which is the smallest convex, closed and solid set containing the set . Kramkov and Schachermayer kram99 , kram03 and Žitković Zit02 , proved that this bipolar set can be characterized as the solid hull of the set , which is defined as the set of supermartingale deflators

Here denotes the set of accumulated gains/losses processes under some admissible portfolios with initial endowment less than or equal to . However, according to the definition of habit formation process , if we derive the naive dual problem using the Legendre–Fenchel transform and the first order condition, we arrive at

The first mathematical difficulty is the extra integral , from which we can see that is not the appropriate space to show the existence of the optimal dual solution. However, it still reminds us to invoke the general treatment of random endowment developed by Cvitanic, Schachermayer and Wang Cvitanic , Karatzas and Žitković KarZit03 and Žitković Zit05 . They proposed another extension of the set , which is now considered as the set of equivalent local martingale measures, to the set of bounded finitely additive measures. Nevertheless, their approach is inadequate to deal with the first term of the dual problem, when the conditional integral part in the conjugate function is taken into account.

In order to avoid the complexity of the path-dependence, we propose the transform from the consumption rate process to the auxiliary process , so that the primal utility maximization problem becomes time separable with respect to the process . This substitution idea from to appeared first in the market isomorphism result for complete markets by Schroder and Skiadas Schroder01072002 . And meanwhile, for each equivalent local martingale measure density process , we define the auxiliary dual process exactly by

The dual problem can therefore be formulated in terms of auxiliary process instead of so that the path dependence of can also be hidden in the definition of process .

By introducing the process , one can shift the integral to the integral with respect to its auxiliary process . With the aid of this equality, we can treat the extra exogenous random term as the shadow random endowment density process and define the dual functional on the properly modified space of instead of . By enlarging the effective domain of values for and , the original utility maximization problem with habit formation can be embedded into the framework of Hugonnier and Kramkov kram04 as an abstract time-separable utility maximization problem on the product space.

On the other hand, we are facing some troubles in applying the classical duality results since the auxiliary process is not integrable. For instance, to show the existence of the dual optimizer, the trick of applying the de la Vallée–Poussin theorem in the proof of Lemma 3.2 in Kramkov and Schachermayer kram99 does not work. And the argument of contradiction in the proof of Lemma in Kramkov and Schachermayer kram03 using the subsequence splitting lemma will also fail by observing that constants may not be contained in the corresponding space. Therefore, we impose the additional sufficient conditions on habit formation discounting factors and ; see Assumption 3.2 to guarantee the well-posedness of the primal optimization problem. We also ask for reasonable asymptotic elasticity conditions on utility functions both at and for the validity of several key assertions of our main results to hold true.

The rest of this paper is organized as follows: Section 2 introduces the financial market and consumption habit formation process. In Section 3, we define the auxiliary process space , the enlarged space and the auxiliary dual space . The original path-dependent utility maximization problem is embedded into an abstract time separable optimization problem with the shadow random endowment. Section 4 is devoted to the formulation of the two-dimensional dual problem over the properly enlarged dual space such that the shadow random endowment part can be hidden, and our main results are stated at the end. Section 5 contains detailed proofs of our main theorems.

## 2 Market model

### 2.1 The financial market model

We consider a financial market with risky assets modeled by a -dimensional semimartingale

(1) |

on a given filtered probability space , where the filtration satisfies the usual conditions, and the maturity time is given by . To simplify the notation, we take . We make the standard assumption that there exists one riskless bond , , which amounts to considering as the numéraire asset.

The portfolio process is a predictable -integrable process representing the number of shares of each risky asset held by the investor at time . The accumulated gains/losses process of the investor under his trading strategy by time is given by

(2) |

### 2.2 Admissible portfolios and consumption habit formation

The portfolio process is called admissible if there exists a constant bound such that , a.s. for all .

Now, given the initial wealth , the agent will also choose an intermediate consumption plan during the whole investment horizon, and we denote the consumption rate process by . The resulting self-financing wealth process is given by

(3) |

Besides of the wealth process, as we defined in the Introduction, the associated consumption habit formation process is given equivalently by the following exponentially weighted average of agent’s past consumption integral and the initial habit:

(4) |

Here discounting factors and measure, respectively, the persistence of the initial habits level and the intensity of consumption history. In this paper, we shall be mostly interested in the general case when discounting factors and are stochastic processes which are allowed to be unbounded. However, for technical reasons, we will assume that a.s. for each .

Throughout this paper, we make the assumption that the consumption habit is addictive, that is, , , which is to say, the investor’s current consumption rate shall never fall below the standard of living process.

A consumption process is defined to be -financeable if there exists an admissible portfolio process such that , a.s. for and the addictive habit formation constraint , a.s. for holds. The class of all -financeable consumption rate processes will be denoted by , for , .

### 2.3 Absence of arbitrage

A probability measure is called the equivalent local martingale measure if it is equivalent to and if is a local martingale under . We denote by the family of equivalent local martingale measures, and in order to rule out the arbitrage opportunities in the market, we assume that

(5) |

See Delbaen and Schachermayer Schachermayer94 and schachermayer98 for comprehensive discussions on the topic of no arbitrage.

Define the RCLL process by

for the . Then is called an equivalent local martingale measure density and with a slight abuse of notation, we denote also as the set of all equivalent local martingale density processes.

The celebrated optional decomposition theorem (see Kramkov MR1402653 ) enables us to characterize the -financeable consumption process by the following budget constraint condition:

###### Proposition 2.1

The process is -financeable if and only if , and

(6) |

### 2.4 The utility function

The individual investor’s preference is represented by a utility function , such that, for every , is continuous on , and for every , the function is strictly concave, strictly increasing, continuously differentiable and satisfies the Inada conditions,

(7) |

where . For each , we extend the definition of the utility function by for all , which is equivalent to the addictive habit formation constraint when the utility function is defined on the difference between the consumption rate process and the habit formation process .

According to these assumptions, the inverse of the function exists for every , and is continuous and strictly decreasing. The convex conjugate of the utility function is defined by

Following the asymptotic elasticity condition on utility functions coined by Kramkov and Schachermayer kram99 (see also Karatzas and Žitković KarZit03 ), we make assumptions on the asymptotic behavior of at both and for future purposes.

The utility function satisfies the reasonable asymptotic elasticity condition both at and , that is,

(8) |

and

(9) |

Moreover, in order to get some inequalities uniformly in time , we shall assume

(10) |

and

(11) |

Many well-known utility functions satisfy reasonable asymptotic elasticity conditions (8) and (9), for example, the discounted log utility function or discounted power utility function ( and ), for some constant . However, it is also easy to check that the utility function does not satisfy condition (9), and the utility function does not satisfy the condition (8).

If the utility function satisfies the lower bound assumption , then condition (9) is automatically verified, and if the utility function satisfies the upper bound assumption , condition (8) holds true.

Next, some technical results give the equivalent characterizations of reasonable asymptotic elasticity conditions (8) and (9). The proof is based on the fact that is a concave function and on similar arguments in Lemma of Kramkov and Schachermayer kram99 ; see also Proposition of Karatzas and Žitković KarZit03 .

###### Lemma 2.1

Let be a utility function satisfying Assumption 2.4. We have if and only if , where we define

(12) |

In each of the subsequent assertions, the infimum of , for which these assertions hold true, equals the reasonable asymptotic elasticity , and the infimum of equals the reasonable asymptotic elasticity . {longlist}[(iii)]

There are and for all s.t.

There are and for all s.t.

There are and for all s.t.

There are and for all s.t.

## 3 A new characterization of financeable consumption processes

### 3.1 Functional set up

In the spirit of Bouchard and Pham pham who treated the wealth dependent problem (see also Žitković Zit05 on consumption and endowment with stochastic clock), we denote as -algebra of optional sets relative to the filtration , and let be the measure on the product space defined as

(13) |

We denote by ( for short) the set of all random variables on the product space with respect to the optional -algebra endowed with the topology of convergence in measure . And from now on, we shall identify the optional stochastic process with the random variable . We also define the positive orthant ( for short) as the set of such that

Endow with the bilinear form valued in as

### 3.2 Path-dependence reduction by auxiliary processes

At this point, we are able to define the set of all -financeable consumption rate processes as a set of random variables on the product space , and Proposition 2.1 states that

(14) |

However, the family may be empty for some values , . We shall restrict ourselves to the effective domain which is defined as the union of the interior of set such that is not empty, and the boundary

(15) |

From the definition, includes the special case of zero initial habit, that is, .

Before we state the next result, we shall first impose some additional conditions on the discounting factors and , which are essential for the well-posedness of the primal utility optimization problem:

We assume that nonnegative optional processes and satisfy

(16) |

and there exists a constant such that

(17) |

If stochastic discounting processes and are assumed to be bounded, conditions (16) and (17) will be satisfied. Condition (16) is the well-known super-hedging property of the random variable in the original market.

###### Lemma 3.1

Under condition (16), the effective domain can be rewritten explicitly as

(18) |

The proof of the lemma is straightforward, and we refer to Yu Yu for details.

By choosing , we can now define the preliminary version of our primal utility maximization problem as

(19) |

Now, for fixed , and each -financeable consumption rate process, we want to generalize the Market Isomorphism idea by Schroder and Skiadas Schroder01072002 in order to reduce the path-dependence structure. By introducing the auxiliary process , the auxiliary set of is given as

(20) |

It is straightforward to verify the following lemma.

###### Lemma 3.2

For each fixed , there is one-to-one correspondence between sets and , and hence we have for .

Let us turn our attention to the set of equivalent local martingale measure densities, and for each , the auxiliary optional process with respect to is defined as

(21) |

Let us denote the set of all these auxiliary optional processes by

We remark here that although stochastic discounting processes and are unbounded, under condition (16), the auxiliary dual process is well defined in , but it is not necessarily in .

A direct application of the Fubini–Tonelli theorem induces the key equalities below; for the detailed proof, we refer to Proposition of Yu Yu .

###### Proposition 3.1

Under condition (16), for each nonnegative optional process such that with defined by (4) for fixed initial standard of living and the nonnegative optional process , we have the following equalities with respect to their corresponding auxiliary processes and which is defined by (21), that

Here we define random processes by

(24) |

Based on Propositions 2.1 and 3.1, under conditions (16) and (17), clearly we will have the alternative budget constraint characterization of the consumption rate process as:

###### Proposition 3.2

For any given pair , the consumption rate process is -financeable if and only if , and

Proposition 3.2 provides us the alternative definition of set for by

(25) |

### 3.3 Embedding into an abstract utility maximization problem with the shadow random endowment

In order to apply the convex duality approach for the random endowment, we need to enlarge the domain of the set to and also enlarge the corresponding auxiliary set to ,

(26) |

where now and is restricted in the enlarged domain ,

Under condition (16) and Proposition 3.1, it is easy to verify the equivalent characterization of by the following:

###### Lemma 3.3

Here

(28) |

and

(29) |

where , and is a well-defined convex cone in . Moreover,

where denotes the closure of the set in .

We will now define the auxiliary primal utility maximization problem based on the auxiliary domain as

(31) |

By definitions of for and for , we successfully embedded our original utility maximization problem (19) with consumption habit formation into the auxiliary abstract utility maximization problem (31) without habit formation, but with the shadow random endowment. More precisely, the following equivalence can be guaranteed that for any ,

(32) |

and the two value functions coincide,

(33) |

In addition, we have that is the optimal solution for if and only if , and for all is the optimal solution for , when .

## 4 The dual optimization problem and main results

Inspired by the idea in Hugonnier and Kramkov kram04 for optimal investment with random endowment, we concentrate now on the construction of the dual problem by first introducing the set ,

(34) |

Let us make the following assumption on stochastic discounting processes and .

The random variable defined by

(35) |

is not replicable under our original financial market; that is, there is no constant such that

Under Assumption 4, the existence of market isomorphism by Schroder and Skiadas Schroder01072002 may no longer hold. Our work can generally extend their conclusions and provide the existence and uniqueness of the optimal solution in incomplete markets using convex analysis.

###### Lemma 4.1

For an arbitrary pair , we denote by the set of nonnegative processes as a proper extension of the auxiliary set in the way that

(37) |

The auxiliary dual utility maximization problem to (31) can be now defined by

(38) |

The following theorems constitute our main results, and we provide their proofs through a number of auxiliary results in the next section.

###### Theorem 4.1

Given Assumptions 3.2 and 4, assume also that conditions (5), (7), (9), (i.e., ), (10) and (11) hold true. Moreover, assume that

(39) |

then we have: {longlist}[(ii)]

The function is -valued on and is -valued on . For each there exists a constant such that , and the conjugate duality of value functions and holds

The solution to the optimization problem (38) exists and is unique (in the sense of under in ) for all such that .

###### Theorem 4.2

In addition to assumptions of Theorem 4.1, we also assume that condition (8) holds, (i.e., ). Then we also have: {longlist}[(iii)]

The value function is -valued on , and is continuously differentiable on .

The solution to optimization problem (31) exists and is unique (in the sense of under in ) for any , and there exists a representation of the optimal solution such that -a.s. for all .

The superdifferential of maps into , that is,

(40) |

Moreover, if , then there exists a representation of the optimal solution such that , -a.s. for all and and are related by

(41) | |||||

If we restrict the choice of initial wealth and initial standard of living such that , the solution to our primal utility optimization problem (19) exists and is unique, moreover,

(42) |

## 5 Proofs of main results

### 5.1 The proof of Theorem 4.1

The following proposition will serve as the key step to build Bipolar relationships:

###### Proposition 5.1

Assume that all conditions of Theorem 4.1 hold true. The families and have the following properties: {longlist}[(ii)]

For any , the set contains a strictly positive random variable on the product space. A nonnegative random variable belongs to if and only if

(43) |

For any , the set contains a strictly positive random variable on the product space. A nonnegative random variable belongs to if and only if

(44) |

In order to prove Proposition 5.1, for any , we denote by the subset of that consists of densities such that . For any density process , define the auxiliary set as