Time-consistent investment and consumption strategies under a general discount function
I. Alia, F. Chighoub, N. Khelfallah, J. Vives
TL;DR
This paper investigates time-consistent investment and consumption strategies under general discount functions using equilibrium policies, addressing time-inconsistency in non-exponential discounting within a stochastic control framework.
Contribution
It extends the concept of equilibrium policies to the Merton problem with non-exponential discounting, providing explicit solutions for common utility functions.
Findings
Characterized equilibrium policies via a stochastic system of forward-backward equations.
Derived explicit equilibrium strategies for power, logarithmic, and exponential utilities.
Addressed time-inconsistency issues in stochastic portfolio optimization.
Abstract
The paper [12] examines a concept of equilibrium policies instead of optimal controls in stochastic optimization to analyze a mean-variance portfolio selection problem. We follow the same approach in order to investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. Equilibrium policies are characterized in this context by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.
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Taxonomy
TopicsStochastic processes and financial applications · Economic theories and models · Risk and Portfolio Optimization
Time-consistent investment and consumption strategies under a general discount function
I. Alia Laboratory of Applied Mathematics, University Mohamed Khider, Po. Box 145 Biskra (07000), Algeria. E-mail: [email protected]
F. Chighoub Laboratory of Applied Mathematics, University Mohamed Khider, Po. Box 145 Biskra (07000), Algeria. E-mail: [email protected]; [email protected]
N. Khelfallah Laboratory of Applied Mathematics, University Mohamed Khider, Po. Box 145 Biskra (07000), Algeria. E-mail: [email protected]
J. Vives Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail: [email protected]
Abstract
In the present paper we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. We consider equilibrium policies within the class of open-loop controls, that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.
Keys words: Stochastic Optimization, Investment-Consumption Problem, Merton Portfolio Problem, Non-Exponential Discounting, Time Inconsistency, Equilibrium Strategies, Stochastic Maximum Principle.
MSC 2010 subject classifications, 93E20, 60H30, 93E99, 60H10.
1 Introduction
Background
The common assumption in classical investment-consumption problems under discounted utility is that the discount rate is assumed to be constant over time which leads to the discount function be exponential. This assumption provides the possibility to compare outcomes occurring at different times by discounting future utility by some constant factor. But on the other hand, results from experimental studies contradict this assumption indicating that discount rates for the near future are much lower than discount rates for the time further away in future. Ainslie, in [1], established experimental studies on human and animal behaviour and found that discount functions are almost hyperbolic, that is, they decrease like a negative power of time rather than an exponential. Loewenstein & Prelec in [19] show that economic decision makers are impatient about choices in the short term but are more patient when choosing between long-term alternatives, and therefore, a hyperbolic type discount function would be more realistic.
Unfortunately, as soon as a discount function is non-exponential, discounted utility models become time-inconsistent in the sense that they do not admit the Bellman’s optimality principle. Consequently, the classical dynamic programming approach may not be applied to solve these problems. In light of the non-applicability of dynamic programming approach directly, there are two basic ways of handling time inconsistency in non exponential discounted utility models. In the first one, under the notion of naive agents, every decision is taken without taking into account that their preferences will change in the near future. The agent at time will solve the problem as a standard optimal control problem with initial condition If we suppose that the naive agent at time [math] solves the problem, his ot her solution corresponds to the so-called pre-commitment solution, in the sense that it is optimal as long as the agent can pre-commit his or her future behavior at time . Kydland & Prescott in [18] indeed argue that a pre-committed strategy may be economically meaningful in certain circumstances. The second approach consists in the formulation of a time-inconsistent decision problem as a non cooperative game between incarnations of the decision maker at different instants of time. Nash equilibrium of these strategies are then considered to define the new concept of solution of the original problem. Strotz in [27] was the first who proposed a game theoretic formulation to handle the dynamic time inconsistent optimal decision problem on the deterministic Ramsey problem, see [26]. Then by capturing the idea of non-commitment, by letting the commitment period being infinitesimally small, he provided a primitive notion of Nash equilibrium strategy. Further work along this line in continuous and discrete time had been done by Pollak [25], Phelps and Pollak [23], Goldman [11], Barro [2] and Krusell & Smith [17]. Keeping the same game theoretic approach, Ekland & Lazrak [7] and Marín-Solano & Navas [20] treated the optimal consumption problem where the utility involves a non-exponential discount function in the deterministic framework. They characterized the equilibrium strategies by a value function which must satisfy a certain ”extended HJB equation”, which is a non linear differential equation displaying a non local term, a term which depends on the global behaviour of the solution. In this situation, every decision at time is taken by a agent which represents the incarnation of the controller at time and is referred in [20] as a ”sophisticated agent”.
Björk & Murgoci in [4] extends the idea to the stochastic setting where the controlled dynamic is driven by a quite general class of Markov process and a fairly general objective function. Yong in [28], by a discretization of time, studied a class of time inconsistent deterministic linear quadratic models and derive equilibrium controls via some class of Riccati-Voltera equations. Yong in [29], also by a discretization of time, investigated a general discounting time inconsistent stochastic optimal control problem and characterizes a feedback time-consistent Nash equilibrium control via the so-called ”equilibrium HJB equation”. In a series of papers, Basak & Chabakauri [3], Hu et al. [13], Czichowsky [6] and Björk et al. [5] look at the mean variance problem which is also time inconsistent.
Concerning equilibrium strategies for an optimal consumption-investment problem with a general discount function, Ekeland & Pirvu [8] are the first to investigate Nash equilibrium strategies where the price process of the risky asset is driven by geometric Brownian motion. They characterize the equilibrium strategies through the solutions of a flow of BSDEs, and they show, for an special form of the discount function, that the BSDEs reduce to a system of two ODEs which has a solution. Ekeland et al. in [9] added life insurance to the investor’s portfolio and they characterize the equilibrium strategy by an integral equation. In [29], Yong discussed the case of time-inconsistent consumption-investment problem under a power utility function. Following Yong’s approach, Zhao et al., in [31], studied the consumption-investment problem with a general discount function and a logarithmic utility function. Recently, Zou et al., in [32], investigated equilibrium consumption-investment decisions for Merton’s portfolio problem with stochastic hyperbolic discounting.
Novelty and Contribution
The purpose of this paper is to investigate equilibrium solutions for a time-inconsistent consumption-investment problem with a non-exponential discount function and a general utility function. Different from [20] and [8] where the authors derived explicit solutions for special forms of the discount factor, in our model, the non-exponential discount function is in a fairly general form. Moreover, we consider equilibrium strategies in the open-loop sense, as defined in [13] and [14], which is different from most of the existing literature on this topic. Note also that the time-inconsistency, in our paper, arises from a non exponential discounting in the objective function, while the works [13] and [14] are concerned with a quite different kind of time-inconsistency which is caused by the presence of non linear term of expectations in the terminal cost. On other hand, the objective functional, in our paper, is not reduced to the quadratic form as in [13] and [14].
By imposing standard assumptions of the classical stochastic maximum principle, we focus on a variational technique approach leading to a version of necessary and sufficient condition for equilibrium, which involves a flow of forward-backward stochastic differential equations (FBSDEs) along with a certain equilibrium condition. We also present a verification theorem that covers some possible examples of utility functions. Then, by decoupling the flow of the FBSDEs, we derive a closed-loop representation of the equilibrium strategies via some parabolic non-linear partial differential equation (PDE). We show that within a special form of the utility function (logarithmic, power and exponential) the PDE reduces to a system of ODEs which has an explicit solution.
We accentuate that, different from most of the existing literature on this topic, where some feedback equilibrium strategies are derived via several very complicated highly non-linear integro-differential equations, an explicit representation of the equilibrium strategies are obtained in our work via simple ODEs. In addition, this method can provide the necessary and sufficient conditions to characterize the equilibrium strategies, while the extended HJB techniques can create, in general, only the sufficient condition in the form of a verification theorem that characterizes the equilibrium strategies.
Structure of the paper
The rest of the paper is organized as follows. In Section 2, we formulate the problem and give the necessary notations and preliminaries. In Section 3 we present the main results of the paper, Theorem 3.2 and Theorem 3.5 that characterizes the equilibrium decisions by some necessary and sufficient conditions. In Section 4, we derive an explicit representation of the equilibrium consumption-investment strategy. Section 5 is devoted to some comparisons with existing results in the literature. The paper ends with an Appendix containing some proofs.
2 Problem formulation
Throughout this paper, will be a filtered probability space such that is a filtration that satisfies the usual conditions, in particular, contains all -null sets and for an arbitrarily fixed finite time horizon Recall that stands for the information available up to time and any decision made at time is based on this information.
We assume that all processes and random variables are well defined and adapted to this filtered probability space. In particular, a dimensional standard Brownian motion
[TABLE]
is defined on
2.1 Notations
Throughout this paper, we use the following notations: : the transpose of the vector (or matrix) , the inner product of and , that is, For a function we denote by (resp. the first (resp. the second) derivative of with respect to the variable
For any Euclidean space with Frobenius norm we let, for any
for any the set of valued measurable random variables such that 2.
the space of valued, adapted processes , with
[TABLE] 3.
the space of valued, adapted continuous processes , with
[TABLE] 4.
the space of valued, adapted processes , with
[TABLE]
2.2 Financial market
Consider an individual facing the inter-temporal consumption and portfolio problem where the market environment consists of one riskless and risky securities. The risky securities are stocks and their prices are modeled as Itô processes. Namely, for the price for of the i-th risky asset, satisfies
[TABLE]
with for and the coefficients and for are progressively measurable processes with values in and , respectively. For brevity, we use to denote the drift rate vector and to denote the random volatility matrix.
The riskless asset, or the savings account, has the price process , for governed by
[TABLE]
where is a deterministic function with values in that represents the interest rate. We assume that for . This is a very natural assumption, since otherwise, nobody is willing to invest in the risky stocks.
2.3 Investment-consumption policies and wealth process
Starting from an initial capital at time [math], during the time horizon , the decision maker is allowed to dynamically invest in the stocks as well as in the bond and consuming. A consumption-investment strategy is described by a -dimensional stochastic process where represents the consumption rate at time and for represents the amount invested in the - risky stock at time The process is called an investment strategy. The amount invested in the bond at time is
[TABLE]
where is the wealth process associated with the strategy and the initial capital . The evolution of can be described as
[TABLE]
Accordingly, the wealth process solves the SDE
[TABLE]
where .
As time evolves, it is natural to consider the controlled stochastic differential equation parametrized by and satisfied by
[TABLE]
Definition 2.1** (Admissible Strategy).**
A strategy is said to be admissible over if and for any the equation has a unique solution
We impose the following assumption about the coefficients.
- (H1)
Processes , and are uniformly bounded. Moreover we assume the following uniform ellipticity condition:
[TABLE]
for some , where denotes the identity matrix on
Under (H1), for any the state equation has a unique solution . Moreover, we have the estimate
[TABLE]
for some positive constant . In particular for and the state equation has a unique solution and the following estimate holds:
[TABLE]
2.4 General discounted utility function
Most of financial-economics works have considered that the rate of time preference is constant (exponential discounting). However there is growing evidence to suggest that this may not be the case. In this section, we discuss the general discounting preferences. We also introduce the basic modeling framework of Merton’s consumption and portfolio problem. We refer the reader to [10], [15], [21], [22] and [24] for more detail about the classical Merton model.
2.4.1 Discount function
As soon as discounting is non-exponential, most papers work with special form of the non-exponential discount factor. Different to these works we consider a general form of the discount factor.
Definition 2.2**.**
A discount function is a deterministic function satisfying and
We also impose the following Lipschitz condition with constant on
- (H2)
There exists a constant such that for any
Remark 2.3**.**
Note that Assumption (H2) is satisfied by many discount functions, such as exponential discount functions, see [21] and [22], mixture of exponential functions, see [8], and hyperbolic discount functions, see [31].
2.4.2 Utility functions and objective
In order to evaluate the performance of a consumption-investment strategy, the decision maker derives utility from inter-temporal consumption and final wealth. Let be the utility of inter-temporal consumption and the utility of the terminal wealth at some non-random horizon (which is a primitive of the model). Then, for any the investment-consumption optimization problem is reduced to maximize the utility function given by
[TABLE]
over subject to where . We restrict ourselves to utility functions which satisfy the following condition:
- (H2)
The maps are strictly increasing, strictly concave and satisfy the Inada conditions.
If we write and we denote and
[TABLE]
then the optimal control problem associated with and is equivalent to maximize
[TABLE]
subject to
[TABLE]
2.4.3 Time inconsistency
Let us first note that the optimal policies, although they exist, will not be time-consistent in general. First of all, as an illustration, let us consider the model in (2.8)–(2.9) with logarithmic utility functions. We suppose that the financial market consists of one riskless asset and risky assets. Arguing as in [8], we can prove that, if the agent is naive and starts with a given positive wealth , at some instant then by the standard dynamic programming approach, the value function associated with this stochastic control problem solves the following Hamilton–Jacobi–Bellman equation
[TABLE]
The HJB equation contains the term , which depends not only on the current time but also on initial time , and so, the optimal policy will depend on as well. Indeed, the first order necessary conditions yield the optimal policy
[TABLE]
Let us consider the following example: The naive agent for the initial pair solves the problem, assuming that the discount rate of time preference will be , for and the optimal consumption strategy will be
[TABLE]
This solution corresponds to the so-called pre-commitment solution, in the sense that it is optimal as long as the agent can precommit (by signing a contract, for example) his or her future behavior at time . If there is no commitment, the 0-agent will take the action but, in the near future, the -agent will change his decision rule (time-inconsistency) to the solution of the HJB equation with In this case the optimal control trajectory for will be changed to given by
[TABLE]
If where is the constant** **discount rate, then
[TABLE]
hence the optimal consumption plan is time consistent. As soon as discount function is non-exponential
[TABLE]
Then the optimal consumption plan is not time consistent. In general, the solution for the naive agent will be constructed by solving the family of HJB equations for , and patching together the “optimal” solutions If the agent is sophisticated, things become more complicated. The standard HJB equation cannot be used to construct the solution, and a new method is required in what follows.
3 Equilibrium strategies
It is well known that the problem described above by turns out to be time inconsistent in the sense that it does not satisfy the Bellman optimality principle, since a restriction of an optimal control for a specific initial pair on a later time interval might not be optimal for that corresponding initial pair. For a more detailed discussion see Ekeland & Pirvu [8] and Yong [29]. Since the lack of time consistency, we consider open-loop Nash equilibrium controls instead of optimal controls. As in [13], we first consider an equilibrium by local spike variation, given, for an admissible consumption-investment strategy For any valued, measurable and bounded random variable and for any define
[TABLE]
We have the following definition.
Definition 3.1** (Open-loop Nash equilibrium).**
An admissible strategy is an open-loop Nash equilibrium strategy if
[TABLE]
for any where is the equilibrium wealth process that solves the SDE
[TABLE]
3.1 A necessary and sufficient condition for equilibrium controls
In this paper we follow an alternative approach, which is essentially a necessary and sufficient conditions for equilibrium. In the same spirit of proving the stochastic Pontryagin’s maximum principle for equilibrium in [13] for the case of linear quadratic models, we derive this condition by a second-order expansion in the spike variation. First, during this section we impose the following hypothesis on the utility functions
- (H3)
The maps are twice continuously differentiable functions. We suppose also that, there exists a positive constant such that
[TABLE]
Now, we introduce the adjoint equations involved in the characterization of open-loop Nash equilibrium controls.
3.1.1 Adjoint processes
Let an admissible strategy and denote by the corresponding wealth process. For each , we introduce the first order adjoint equation defined on the time interval , and satisfied by the pair of processes as follows
[TABLE]
where Under the assumption (H1), the BSDE is uniquely solvable in Moreover there exists a constant such that, for any we have the following estimate
[TABLE]
The second order adjoint equation is defined on the time interval and satisfied by the pair of processes as follows
[TABLE]
where Under (H1) the above BSDE has a unique solution . Moreover we have the following representation for
[TABLE]
Indeed, if we define the function for each as the fundamental solution of the linear ODE
[TABLE]
and we apply the Itô’s formula to on by taking conditional expectations, we obtain . Note that since , then .
3.1.2 A characterization of equilibrium strategies
The following theorem is the first main result of this work, it provides a necessary and sufficient condition for equilibrium. As we have said before, the proof is inspired in [13] and [14].
First, we define the process and we introduce the following notations:
[TABLE]
and
[TABLE]
Theorem 3.2**.**
Let (H1)-(H3) hold. Given an admissible strategy , let for any the processus
[TABLE]
be the unique solution to the BSDE . Then, is an equilibrium consumption-investment strategy, if and only if, the following condition holds
[TABLE]
In order to derive the proof of this theorem, let us, first of all, derive some technical results. First, denote by the solution of the state equation corresponding to . Since the coefficients of the controlled state equation are linear, using the standard perturbation approach, see e.g. [30], we have
[TABLE]
where for any valued, measurable and bounded random variable and for any and solve respectively the following linear stochastic differential equations:
[TABLE]
and
[TABLE]
Proposition 3.3**.**
*Let **(H1) *holds. For any the following estimates hold for any
[TABLE]
In addition, we have the following equality
[TABLE]
Proof.
See the Appendix. ∎
Now, we present the following technical lemma needed later. The proof follows an argument adapted from Hu el al. [14],
Lemma 3.4**.**
Under assumptions (H1)-(H3), the following two statements are equivalent
- i)
* *
- ii)
* *
Proof.
See the Appendix. ∎
Proof of Theorem 3.2. Given an admissible strategy
[TABLE]
for which holds, according to Lemma 3.4, we have, for any
[TABLE]
Then, from for any and for any valued, measurable and bounded random variable
[TABLE]
where we have used in the last inequality the fact that, under the concavity condition of and ,** **it follows Hence is an equilibrium strategy.
Conversely, assume that is an equilibrium strategy. Then, by together with for any the following inequality holds:
[TABLE]
Now, we define
[TABLE]
Clearly is well defined. In fact, it is a second order polynomial in terms of the components of vector Easy manipulations show that the inequality is equivalent to
[TABLE]
And it is easy to see that the maximum condition leads to the following condition:
[TABLE]
According to Lemma 3.4, the expression follows immediately. This completes the proof.
3.2 A characterization of equilibrium strategies by verification
argument
In classical (time-consistent) stochastic control theory the sufficient condition of optimality is of significant importance for computing optimal controls. It says that if an admissible control satisfies the maximum condition of the Hamiltonian function, then the control is indeed optimal for the stochastic control problem. This allows one to solve examples of optimal control problems where one can find, a smooth solution to the associated adjoint equation.
It is worth mentioning also that, the assumption (H3) is too strong to apply the theorem 3.2. to some important problems in the practice. For example, the power utility function does not satisfy this assumption. The aim of the following theorem is to characterize the open-loop equilibrium pair by a sufficient condition of equilibrium. In order to overcome the technical difficulties mentioned by the hypothesis (H3), let us introduce further conditions about the utility functions.
- (H4)
The maps are continuously differentiable and the first order derivatives are continuous.
Theorem 3.5**.**
*Let **(H1), (H2) *and (H4) hold. Given an admissible strategy , let for any the processus
[TABLE]
be the unique solution to the BSDE . Then, is an equilibrium consumption-investment strategy, if the following condition holds
[TABLE]
**Proof. **Suppose that is an admissible control for which the condition (3.22) holds, in addition for any and , we consider by then we have the following difference
[TABLE]
Noting that, by the concavity of , we have
[TABLE]
Accordingly, by the terminal condition in the BSDE (3.4) we obtain that
[TABLE]
By applying Ito’s formula to on , we get
[TABLE]
By the concavity of , we find
[TABLE]
By taking and in it follows that
[TABLE]
Now dividing both sides by and taking the limit when vanishes, by Lemma 3.4, we conclude tha · is an equilibrium control.
Remark 3.6**.**
The purpose of the sufficient condition of optimality is to find an optimal control by computing the difference in terms of the Hamiltonian function, where be an arbitrary admissible control. Here, the spike variation perturbation plays a key role in deriving the sufficient condition for equilibrium strategies, which reduces to the computation of the difference , without the necessity to achieving the second order expansion in the spike variation.
4 Equilibrium when the coefficients are deterministic
Theorem 3.5. shows that one can obtain equilibrium consumption-investment strategies by solving a system of FBSDEs which is not standard since the “flow” of the unknown process is involved. Moreover, there is an additional constraint that act on the “diagonal” (i.e. when ) of the flow. As far as we know, the explicitly solvability of this type of equations remains an open problem, except for some particular form of the utility functions. However, we are able to solve quite thoroughly this problem when the parameters and are deterministic functions. In this section we define what we mean by an equilibrium rule, and then we derive a parabolic backward PDE. Our PDE is comparable with the one obtained in [20] and [8], for some particular discount functions in finite horizon with different utility functions.
In this section, let us look at the Merton’s portfolio problem with general discounting and deterministic parameters. At first, we consider the following parabolic backward partial differential equation
[TABLE]
where we denote by the inverse function of the strictly decreasing marginal derivative utility and .
We have the following verification theorem
Theorem 4.1**.**
*Let **(H1), (H2) and (H4) *hold. If there exists a classical solution
[TABLE]
of the PDE such that the stochastic differential equation
[TABLE]
has a unique solution in which the following estimate holds
[TABLE]
Then, the equilibrium consumption-investment strategy is given by
[TABLE]
Proof. Suppose that is an equilibrium control and denote by the corresponding wealth process. Then, in view of Theorem , there exist an adapted process solution of the following flow of forward-backward SDEs, parametrized by
[TABLE]
with conditions
[TABLE]
From the terminal condition in the first order adjoint process we consider the following Ansatz
[TABLE]
for some deterministic function such that
Applying Itô’s formula to , it yields
[TABLE]
Next, comparing the term in by the ones in the second equation in we deduce that
[TABLE]
and by comparing the terms we also get
[TABLE]
We put the above expressions of and at into and then
[TABLE]
and
[TABLE]
which leads to the following representation
[TABLE]
Then by taking expressions and into , this suggests that coincides with the solution of the PDE , evaluated along the trajectory solution of the state equation.
Remark 4.2**.**
Equation is comparable with the one in Marín-Solano & Navas [20] and Ekland & Pirvu [8], in which the equilibrium is defined within the class of feedback controls.
Remark 4.3**.**
Theorem 4.1. enables us to derive a suitable equilibrium strategie as well as , at each , this permits us to derive directly an explicit expression of equilibrium control in the cases of power, logarithmic and exponential utility functions. While the duality approach [12] permits to characterize a stochastic equilibrium solution in terms of a complicated FBSDE system of a closed form; it does not provide an explicit represntation.
5 Special utility functions
Equilibrium investment-consumption strategies for Merton’s portfolio problem with general discounting and deterministic parameters have been studied in [20], [8] and [29], among others, in different frameworks. In this section, we discuss some special cases in which the function may be separated into functions of time and state variables. Then, one needs only to solve a system of ODEs in order to completely determine the equilibrium strategies. We will compare our results with some existing ones in the literature.
5.1 Power utility function
To make the problem explicitly solvable, we consider power utility functions for the running and terminal costs. That is, and with and In this case the PDE reduces to
[TABLE]
From the terminal condition, we consider the following trial solution
[TABLE]
for some deterministic function with the terminal condition Then by substituting in we obtain
[TABLE]
where
[TABLE]
and
[TABLE]
It remains to determine the function First, by the change of variable
[TABLE]
we find that should solve the following ODE
[TABLE]
A variation of constant formula yields to
[TABLE]
and subsequently we obtain
[TABLE]
In view of Theorem 4.1, the representation of the Nash equilibrium strategies - gives
[TABLE]
This consumption–investment strategy determines a wealth process given by
[TABLE]
The abouve solution is comparable with the one obtained by Marín-Solano & Navas [20], Ekland & Pirvu [8] and Yong [29].
5.2 Logarithmic utility function
Now, let us analyze the case where and with In this case, the PDE reduces to
[TABLE]
Once again, we know that the solution of will be of the form
[TABLE]
where By substituting in we get
[TABLE]
which is explicitly solved by
[TABLE]
In view of Theorem 4.1, the representation of the Nash equilibrium strategies - gives
[TABLE]
This consumption–investment strategy determines a wealth process given by
[TABLE]
5.3 Exponential utility function
Next, we consider the case where and with . The terminal condition PDE becomes
[TABLE]
We try a solution of the form
[TABLE]
where , such that and By substituting in we get
[TABLE]
This suggests that functions and should solve the following system of equations
[TABLE]
which is explicitly solvable for by
[TABLE]
and
[TABLE]
The representation of the Nash equilibrium strategies - gives
[TABLE]
This consumption–investment strategy determines a wealth process given by
[TABLE]
The above solution is comparable with the ones obtained in Marín-Solano & Navas [20] by solving an extended Hamilton–Jacobi–Bellman (HJB) equations.
6 Special discount function
As well documented in [20], an agent making a decision at time is usually called the -agent, and can act in two different ways: naive and sophisticated. Naive agents take decisions without taking into account that their preferences will change in the near future, and then any -agent will solve the problem as a standard optimal control problem with initial condition and his decision will be in general time-inconsistent. In order to obtain a time consistent strategy, the -agent should be sophisticated, in the sense of taking into account the preferences of all the -agents, for . Therefore, the approach to handle the time inconsistency in dynamic decision making problems is by considering time-inconsistent problems as non-cooperative games with a continuous number of players, in which decisions at every instant of time are selected. The solution to the problem of the agent with non-constant discounting should be constructed by looking for the sub-game perfect equilibria of the associated game with an infinite number of -agents. In [20] the authors looked for a solution of a sophisticated agent to the modified HJB (which is not a partial differential equation due to the presence of a non-local term). Then, they need to define the Markov equilibrium strategies, while in our work, and different from [20], we use the open-loop equilibrium strategies. This is a significant difference which leads to obtain an important change in the results.
6.1 Exponential discounting with constant discount rate (classical
model)
At first, we consider the standard exponential discount function , , where is a constant representing the discount rate. In this case, our equilibrium solution for the three cases become
- Logarithmic utility
[TABLE]
- Power utility
[TABLE]
- Exponential utility
[TABLE]
where are given by and respectively, and
[TABLE]
Notice that our solutions given above coincide with the optimal solutions of classical Merton portfolio problem (see e.g.[20] in the case with constant discount rate). This confirms the well-known fact that the time-consistent equilibrium strategy for an exponential discount function is nothing but the optimal strategy. A relevant remark is that the portfolio rule is independent of the discount factor, and it is the same for a non-exponential discount function.
6.2 Exponential discounting with non constant discount rate
(Karp’s model)
Now, following Karp [16], let us assume that the instantaneous discount rate is non-constant, but a continuous and positive function of time , for Impatient agents will be characterized by a non-increasing discount rate . The discount factor used to evaluate a payoff at times 0, is given by
[TABLE]
In this case, the objective is exactly the same as Marín-Solano and Navas [20], in which the equilibrium is however defined within the class of feedback controls. In [20], the (feedback) equilibrium consumption-investment solutions (also called the sophisticated consumption-investment strategies) are summarized as
- Logarithmic utility
[TABLE]
- Power utility
[TABLE]
where is the solution of the integro-differential equation,
[TABLE]
with given by and
[TABLE]
- Exponential utility
[TABLE]
where is given by and satisfies the following very complicated integro-differential equation,
[TABLE]
where
[TABLE]
with
[TABLE]
Our (open-loop) equilibrium solutions reduce to
- Logarithmic utility
[TABLE]
- Power utility
[TABLE]
- Exponential utility
[TABLE]
where are given by and respectively, and
[TABLE]
Remark 6.1**.**
Comparing the results of this special case with our solutions, we find the following facts: The equilibrium proportion investment strategies coincide in the three cases. The consumption strategies are different in the three cases. Moreover, our equilibrium consumption strategies are well defined and explicitly given, while in [20], equilibrium consumption strategies in the case of Power utility as well as in the case of Exponential utility, are obtained via a very complicated integro-differential equations, whose unique solvability are not established.
7 Appendix
Following [13], we derive the proof of Proposition 3.3 by means of the duality analysis. Moreover, since our objective function is not in quadratic form, we need to adapt the results obtained in [13] according to our control problem which concerns a general and non necessary quadratic utility maximization.
Proof of Proposition 3.3. The estimates follow from Theorem 4.4 in [30]. Moreover the following expansion holds for the objective functional
[TABLE]
Now, applying the second order Taylor-Lagrange expansion to we find
[TABLE]
Notice that
[TABLE]
where denote the Euclidean norm of , that is Then we obtain
[TABLE]
Noting that, by the second order Taylor-Lagrange expansion, see e.g. [30], we have for some constant
[TABLE]
then the following expansion holds for the objective functional
[TABLE]
Notice that
[TABLE]
Now, by applying Ito’s formula to on , we get
[TABLE]
Again, by applying Ito’s formula to on we get
[TABLE]
where On the other hand, we conclude from (H1) together with that
[TABLE]
By taking and in it follows that
[TABLE]
which is equivalent to
**Proof of Lemma 3.4. ** We set up
[TABLE]
Now we define, for and
[TABLE]
Then, for any in the interval the pair satisfies
[TABLE]
Moreover, it is clear that from the uniqueness of solutions to , we have the equality for any such that Hence, the solution does not depend on the variable and this allows us to denote the solution of by
We have then, for any and
[TABLE]
Now using we have, under (H2), for any and
[TABLE]
and
[TABLE]
From which, we have for any
[TABLE]
Thus
[TABLE]
From the above equality, it is clear that if (ii) holds, then,
[TABLE]
Conversely, according to Lemma 3.4 in [14], if (i) holds, then,
[TABLE]
This completes the proof.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] G. Ainslie (1995): Special reward: A behavioral theory of impulsiveness and impulse control . Psychological Bulletin 82: 463-496.
- 2[2] R. J. Barro (1990): Ramsey meets Laibson in the neoclassical growth model . Quarterly Journal of Economics 114: 1125-1152.
- 3[3] S. Basak and G. Chabakauri (2010): Dynamic mean-variance asset allocation . Review of Financial Studies 23: 2970-3016.
- 4[4] T. Björk and A. Murgoci (2008): A general theory of Markovian time-inconsistent stochastic control problems . SSRN Electronic Journal 1694759.
- 5[5] T. Björk, A. Murgoci and X. Y. Zhou (2014): Mean-variance portfolio optimization with state-dependent risk aversion . Mathematical Finance 24 (1): 1-24.
- 6[6] C. Czichowsky (2013): Time-consistent mean-variance portfolio selection in discrete and continuous time . Finance and Stochastics 17 (2): 227-271.
- 7[7] I. Ekeland and A. Lazrak (2008): Equilibrium policies when preferences are time inconsistent . Ar Xiv:0808.3790 v 1.
- 8[8] I. Ekeland and T. A. Pirvu (2008): Investment and consumption without commitment . Mathematics and Financial Economics 2: 57-86.