Recursive Utility Maximization for Terminal Wealth under Partial Information

This paper concerns the recursive utility maximization problem for terminal wealth under partial information. We first transform our problem under partial information into the one under full information. When the generator of the recursive utility is concave, we adopt the variational formulation of the recursive utility which leads to a stochastic game problem and characterization of the saddle point of the game is obtained. Then, we study the K-ignorance case and explicit saddle points of several examples are obtained. At last, when the generator of the recursive utility is smooth, we employ the terminal perturbationmethod to characterize the optimal terminal wealth.


Introduction
In this paper, we study the problem of an agent who invests in a financial market so as to maximize the recursive utility of his terminal wealth () on finite time interval [0, ], while the recursive utility is characterized by the initial value (0) of the following Backward stochastic differential equation ( (1) The market consists of a riskless asset and  risky assets, the latter being driven by a -dimensional Brownian motion.And the investor has access only to the history of interest rates and prices of risky assets, while the appreciation rate and the driving Brownian motion are not directly observed.That is, the filtration generated by the Brownian motion could not be used when the investor chooses his portfolios.This is quite practical in a real financial market.So we are interested in this so-called recursive utility maximization problem under partial information.
In the full information case, the problem of maximizing the expected utility of terminal wealth is well understood in a complete or constrained financial market [1,2].In an incomplete multiple-priors model, Quenez [3] studied the problem of maximization of utility of terminal wealth in which the asset prices are semimartingales.Schied [4] studied the robust utility maximization problem in a complete market under the existence of a "least favorable measure."As for the recursive utility optimization, El Karoui et al. [5] studied the optimization of recursive utilities when the generator of BSDE is smooth.Epstein and Ji [6,7] formulated a model of recursive utility that captures the decision-maker's concern with ambiguity about both the drift and ambiguity and studied the recursive utility optimization under -framework.Hu et al. [8] introduced a BSDE driven by -Brownian motion from which a kind of more general recursive utility can be defined.Then Hu and Ji [9,10] studied the corresponding control problem by two methods: maximum principle and dynamic programming principle.But all the above works do not accommodate partial information.
In the partial information case, Lakner [11] generalized the martingale method to expected utility maximization problem; see also Pham [12].Cvitanić et al. [13] maximized the recursive utility under partial information.But the generator  in Cvitanić et al. [13] does not depend on .Miao [14] studied a special case of recursive multiple-priors utility maximization problem under partial information in which the appreciation rate is assumed to be an F 0 -measurable, unobserved random variable with known distribution.Actually, they studied the problem under Bayesian framework and did not give the explicit solutions.
In this paper, we first transform our portfolio selection problem under partial information into one under full information in which the unknown appreciation rate is replaced by its filter estimate and the Brownian motion is replaced by the innovation process.Then a backward formulation of the problem under full information is built in which instead of the portfolio process, the terminal wealth is regarded as the control variable.This backward formulation is based on the existence and uniqueness theorem of BSDE and was introduced in [5,15].
When the generator  of ( 1) is concave, we adopt the variational formulation of the recursive utility which leads to a stochastic game problem.Inspired by the convexity duality method developed in Cvitanić and Karatzas [16], we turn the primal "sup-inf" problem to a dual minimization problem over a set of discounting factors and equivalent probability measures.A characterization of the saddle point of this game is obtained in this paper.Furthermore, the explicit saddle points for several classical examples are worked out.
When the generator  of the BSDE is smooth, we apply the terminal perturbation method developed in Ji and Zhou [17] and Ji and Peng [18] to characterize the optimal terminal wealth of the investor.Once the optimal terminal wealth is obtained, the determination of the optimal portfolio process is a martingale representation problem which we do not involve in this paper.
The rest of this paper is organized as follows.In Section 2, we formulate the recursive utility maximization problem under partial information, reduce the original problem to a problem under full information, and give the backward formulation.The case of nonsmooth generator is tackled in Section 3. In Section 4, we specialize in -ignorance model and give explicit saddle points of several examples.In Section 5, we characterize the optimal wealth when the generator  is smooth.
The asset prices are assumed to be continuously observed by the investors in this market; in other words, the information available to the investors is represented by G = {G  } ≥0 , which is the -augmentation of the filtration generated by the price processes ((); 0 ≤  ≤ ).The matrix disperse coefficient () is assumed invertible, bounded uniformly, and ∃ > 0,   ()  () ≥ ‖‖ 2 , ∀ ∈ R  ,  ∈ [0, ], a.s.In fact, () can be obtained from the quadratic variation of the price process.So we assume w.l.o.g. that () is G  -adapted.However, the appreciation rate   () fl ( 1 (), . . .,   ()) is not observable for the investors.
A small investor whose actions cannot affect the market prices can decide at time  ∈ [0, ] what amount   () of his wealth to invest in the th stock,  = 1, . . ., .Of course, his decision can only be based on the available information {G  }  =0 ; that is, the processes Then the wealth process (⋅) ≡  , (⋅) of a self-financing investor who is endowed with initial wealth  > 0 satisfies the following stochastic differential equation: =   ()  ()  +   ()  ()  () . ( Because the only information available to the investor is G, we could not use the Brownian motion  to define the recursive utility.As we will show in the following, there exists a Brownian motion Ŵ under  in the filtered measurable space (Ω, G) which is often referred to as an innovation process.The recursive utility process () ≡  , () of the investor is defined by the following backward stochastic differential equation: where  and  are functions satisfying the following assumptions.
Assumption 2.  : R + → R is continuously differentiable and satisfies linear growth condition.Remark 3. Equation ( 4) is not a standard BSDE because in general G is strictly larger than the augmented filtration of the (, G)-Brownian motion Ŵ.
We will show in the next subsection that under Assumption 1, for any  ∈  2 G  , the BSDE (4) has a unique solution G T , and Assumption 2 ensures that the variable (()) ∈  2 G  .Thus, under Assumptions 1 and 2, the recursive utility process associated with this terminal value is well defined.
Given initial wealth  > 0, denote by A() the set of investor's feasible portfolio strategies; that is

Reduction to a Problem under Full Information.
Define the risk premium process () = () −1 ().Because we have assumed the processes (⋅) and (⋅) are uniformly bounded, the process is a (, F) martingale.So a probability measure P can be defined by ∀ ∈ F  , where P is usually called risk neutral probability in the financial market.The process is a Brownian motion under P by Girsanov's theorem.
Let η() fl [() | G  ] be a measurable version of the conditional expectation of  with respect to the filtration G.
We introduce the process By Theorem 8.1.3and Remark 8.1.1 in Kallianpur [20], { Ŵ(),  ≥ 0} is a (G, )-Brownian motion which is the so-called innovations process.Then, we could describe the dynamics of stock price processes and the wealth process within a full observation model:

Backward Formulation of the Problem.
In this subsection, we first show BSDE (4) has a unique solution under some mild conditions and then give an equivalent backward formulation of problem (15).
Since G is strictly larger than the augmented filtration of the (, G)-Brownian motion Ŵ in general, equation ( 4) is not a standard BSDE.Fortunately, by Theorem 8.3.1 in [20], every square integrable G  -martingale () can be represented as where (⋅) ∈  2 G .Thus, applying similar analysis as in [21], it is easy to prove this lemma.
It is clear that original problems ( 7) and ( 15) are equivalent to the auxiliary one (19).Hence, hereafter we focus ourselves on solving (19).Note that  becomes the control variable.The advantage of this approach is that the state constraint in (7) becomes a control constraint in (19), whereas it is well known in control theory that a control constraint is much easier to deal with than a state constraint.The cost of this approach is the original initial condition   (0) =  that now becomes a constraint.
Feasible  * ∈ A() is called optimal if it attains the maximum of () over A().Once  * is determined, the optimal portfolio can be obtained by solving the first equation in (17) with   * () =  * .

Dual Method for Recursive Utility Maximization
In this section, we impose the following concavity condition.We also need the following assumption on .

𝐸 [ [
where the last inequality is due to (33).
In the following, we prove the existence of the quadruple ( ξ, β, γ, ζ) which is postulated in Lemma 11.Notice that the function   → ũ(1/) is convex.By similar analysis as in Appendix B of [23], the following lemma holds.
Our main result is the following theorem.
Remark 17.It is worth pointing out that the adjoint process   in the proof of the above theorem coincides with the optimal utility process   in (55).

𝐾-Ignorance
In this section, we study a special case which is called -ignorance by Chen and Epstein [25].In this case, the generator  is specified as Chen and Epstein interpreted the term || as modeling ambiguity aversion rather than risk aversion.() = −|| is not differentiable.But it is concave and () = inf ||≤ ().
Then our results in the above section are still applicable.
In this section, we assume  = 1,  ≡ 1.The wealth equation and recursive utility become  (63) Now Lemma 9 can be simplified to the following lemma.Lemma 18.For  ∈ , the solutions ((⋅), (⋅)) and ((⋅), (⋅)) of ( 62) can be represented as where Ẽ[⋅] is the expectation operator with respect to the risk neutral measure P. By Theorem 2.   The uniqueness follows from the strict convexity of .This completes the proof.