Recursive utility optimization with concave coefficients

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal"sup-inf"problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions.

But up to our knowledge, there are not many results related to this kind of optimization problem with nonlinear wealth equations. Cvitanic and Cuoco [3] explored the optimal consumption problem for a large investor whose portfolio strategies can affect the instantaneous expected returns of the assets. They show the existence of optimal policies by convex duality method developed in [4,19]. Ji and Peng [14] studied the continuous time mean-variance problem with nonlinear wealth equation. El Karoui et al [9] obtained a dynamic maximum principle for the optimization of recursive utilities and characterized the optimal consumptions and portfolio strategies via a forward-backward SDE system. Their method depends heavily on the smoothness of the coefficients of the forward-backward SDE system.
As for the wealth equations, there are some interesting cases in which the coefficients are nonlinear and nonsmooth. As shown in [3], the wealth equation of a large investor may be nonsmooth. The other well-known case is that an investor is allowed to borrow money with a higher interest rate. As for the recursive utilities, some important generators are also nonsmooth, for instance, the K-ignorance case which was proposed by Chen and Epstein [2]. The coefficients of the wealth equations and the recursive utilities of the above cases are all concave. This motivate us to study the recursive utility maximization problem with concave coefficients of both the wealth equations and the recursive utilities in this paper.
We first give an equivalent backward formulation of our problem. This "backward formulation" was introduced by El Karoui, Peng and Quenez [9] in order to solve a recursive utility optimization problem, and employed by Ji and Peng [14] to obtain a necessary condition for a mean-variance portfolio selection problem with non-convex wealth equations. For its application in stochastic control with state constraints, we refer the reader to Ji, Zhou [15,16]. El Karoui, Peng and Quenez [9] took the terminal wealth as the control variable, and then used a variational technique to obtain a stochastic maximum principle, i.e., a firstorder necessary condition that characterizes the terminal wealth. In our context, we still take the terminal wealth as the control variable. But the stochastic maximum principle approach does not work due to the nonsmoothness of the coefficients. In order to overcome this difficulty, we assume that the coefficients are concave and derive a variational formulation by the Fenchel-Legendre transform of the coefficients which leads to a stochastic game problem. Inspired by the convex duality method developed in Cvitanic and Karatzas [5], we turn the primal "sup-inf" problem to a dual minimization problem and the saddle point of this game is derived. Then we obtain the optimal terminal wealth and the optimal portfolio process can be derived by the martingale representation theorem.
Three cases for investors with ambiguity aversion are provided to show the applications of our results.
In these cases, we specialize the generator of the recursive utility as the K-ignorance case and the utility function of the terminal wealth as u(x) = 1 α x α , 0 < α < 1. By the main results in section 4, we characterize the saddle point via a quadratic BSDE and obtain the optimal terminal wealth explicitly. Especially, for the large investor case, we work out the explicit saddle point, the optimal wealth process, the optimal portfolio strategies as well as the utility intensity process. This paper is organized as follows. In section 2, we give the classical, backward and variational formulation of the recursive utility maximization problem. Our main results are obtained in section 3. In section 4, we study three cases in which the investors are assumed to be ambiguity aversion (K-ignorance). The saddle point and the optimal terminal wealth for each case are derived explicitly.

Formulation of the problem
In this paper, we study the recursive utility maximization problem with bankruptcy prohibition.

The wealth process
Let W = (W 1 , ..., W d ) ′ be a standard d-dimensional Brownian motion defined on a filtered complete probability space (Ω, F , {F t } t≥0 , P ), where {F t } t≥0 denotes the natural filtration associated with the d-dimensional Brownian motion W and augmented.
Consider a financial market consisting of a riskless asset (the money market instrument or bond) whose price is S 0 and d risky securities (the stocks) whose prices are S 1 , ..., S d . An investor decides at time t ∈ [0, T ] what amount π i t of his wealth X t to invest in the ith stock, i = 1, ..., d. The portfolio π t = (π 1 t , ..., π d t ) ′ and We suppose that the wealth process X t ≡ X x,π t of the investor who is endowed with initial wealth x > 0 is governed by the following stochastic differential equation, where b is a given function and the predictable and invertible process σ t = {σ ij t } 1≤i,j≤d is the stock volatility. σ t is assumed to be bounded, uniformly in (t, ω) The price processes S 0 t and S 1 t , ..., S d t are governed by are assumed to be predictable and bounded, uniformly in (t, ω) ∈ [0, T ]×Ω. Then the wealth process X t satisfies the following linear stochastic differential where 1 is the d-dimensional vector whose every component is 1.
The borrowing rate R t is higher than the risk-free rate r t .
The stock prices are (2.2). Now the borrowing rate R t is higher than the risk-free rate r t , i.e., R t ≥ r t , t ∈ [0, T ], a.s. In this case, the wealth process becomes Cuoco and Cvitanic [3] considered the optimal portfolio choice problem for a large investor whose portfolio strategies can affect the price processes of the securities. In [3], the price processes are given by where l i : R + × R → R, i = 0, 1, ..., d are some given functions which describe the effect of the wealth and the strategies possessed by the large investor. The wealth process is governed by Cuoco and Cvitanic [3] also gave the following more specific example. For x ∈ R, (2.5) Set ε = (ε 1 , ..., ε d ) ′ , sgn(π t ) = (sgn(π 1 t ), ..., sgn(π d t )) ′ and ε ⊗ sgn(π t ) = (ε 1 sgn(π 1 t ), ..., ε d sgn(π d t )) ′ where ε i are given small positive numbers.
Consider l 0 (X t , π t ) = 0 and l i (X t , π t ) = −ε i sgn(π i t ), i = 1, ..., d. Then we have For this specific large investor model, longing the ith risky security depresses its expected return while shorting it increases its expected return as explained in Cuoco and Cvitanic [3].
We introduce the following spaces: For notational simplicity, we will often write L 2 , M 2 and S 2 instead of L 2 (Ω, F T , P ; R), M 2 (0, T ; R d ) and S 2 (0, T ; R) respectively.

The recursive utility
In the time-additive expected utility maximization models, one can not separate the risk aversion and intertemporal substitution. To overcome this intertwine, Duffie and Epstein [6] introduced the stochastic recursive utility in the continuous time. In El Karoui, Peng and Qeuenz [10], the stochastic recursive utility can be formulated in a more general form by backward stochastic differential equation (BSDE for short): We need the following assumptions.
and satisfies (i) There exists a constant C ≥ 0 such that Assumption 2.6 u : (0, ∞) → R is strictly increasing, strictly concave and of class C 2 , and satisfies (i) Inada condition:

Classical formulation
We consider that an investor chooses a portfolio strategy so as to where X t ≥ 0 describes that no-bankruptcy is required.
Definition 2.7 A portfolio π is said to be admissible if π ∈ M 2 and the corresponding wealth process Given the initial wealth x > 0, denote byĀ(x) the set of an investor's admissible portfolio strategies, Thus, (2.8) can be written as:

Backward formulation
In this subsection, we give an equivalent backward formulation of the above optimization problem (2.8).
Since σ t is invertible, q t can be regarded as the control variable instead of π t . Notice that selecting q is equivalent to selecting the terminal wealth X T by the existence and uniqueness result of BSDEs (refer to Theorem 2.1 in [10]). Hence the wealth equation (2.1) can be rewritten as where the terminal wealth ξ is the "control" to be chosen from U . Note that we will require that the solution X of (2.10) at time 0 equals the initial wealth x.
If we take the terminal wealth as control variable, the recursive utility process can be written as: (2.11) Assumption 2.6 guarantees that u(ξ) ∈ L 2 for any ξ ∈ U . By the existence and uniqueness result of BSDEs, we know that for any ξ ∈ U , there exists a unique solution (X t , q t ) (resp. (Y t , Z t )) of (2.10) (resp. (2.11)). By the comparison theorem of BSDEs, Assumption 2.4 and the nonnegative terminal wealth keeps the wealth process be nonnegative all the time. Usually, we denote the solution Y of (2.11) at tome 0 by Y ξ 0 . This gives rise to the following optimization problem: (X, q) and (Y, Z) satisfy (2.10) and (2.11) respectively. (2.12) Definition 2.8 A random variable ξ ∈ U is called feasible for the initial wealth x if and only if X 0 = x.
We will denote by A(x) the set of all feasible ξ for the initial wealth x.
It is clear that the original problem (2.8) is equivalent to the auxiliary one (2.12). Hence, hereafter we focus ourselves on solving problem (2.12). The advantage of doing this is that, since ξ is the control variable, the state constraint in (2.8) becomes a control constraint in (2.12), 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 that the original initial condition X 0 = x becomes a constraint.

Variational formulation
The effective domain ofb is As was shown in [10], the (ω, t)-section of Db, denoted by D is included in the bounded domain where C 1 is the Lipschitz constant of b. The following duality relation is due to the concavity of b, Let F (ω, t, β, γ) be the Fenchel-Legendre transform of f : The effective domain of F is where C is the Lipschitz constant of f . We have the duality relation by the concavity of f , For and denote by (X µ,ν , q µ,ν ) the unique solution to the linear BSDE (2.10) associated to b µ,ν . For any and denote by (Y β,γ , Z β,γ ) the unique solution to the linear BSDE (2.11) associated to f β,γ .
By the method similar to Proposition 3.4 in [10], we have the following variational formulation of X t and Y t .
Under Assumption 2.4 and 2.5, the solutions (X t , q t ) and (Y t , Z t ) of (2.10) and (2.11) can be represented Especially, we have By the above analysis, Now our problem (2.12) is equivalent to the following problem: It is essentially a robust optimization problem. Define the "max-min" quantity It is dominated by its "min-max" counterpart If we can find (β,γ,ξ) ∈ B × A(x) such that then the optimal solution of problem (2.18) isξ.
By Assumption 2.9, there exists a constant M 1 such that ∀(β, γ) ∈ B, Due to the monotonicity ofũ and the convexity of (x, y) → xũ(ζ y x ), we deduce the following inequality by Jensen's inequality where the constants M 2 > 0, M 3 > 0 depend on the bound of F and the Lipschitz constants of b, f . Then we have So there exists a numberζ x ∈ (0, ∞) which attains the infimum of V * (x).
This completes the proof.
Proof: The proof is divided into three steps.
This completes the proof.

Investors with ambiguity aversion
We model the utility process (2.7) via where K is a given d-dimensional vector whose component K i ≥ 0, i = 1, 2, ..., d. Chen and Epstein [2] interpreted the term −K ′ |z| as modeling ambiguity aversion rather than risk aversion. This special formulation in [2] is called K-ignorance. When K i = 0, i = 1, 2, ..., d, it degenerates to the classical expected utility maximization problem studied in [3], [19] etc.
For the K-ignorance case, we have For a given γ ∈ B 2 , define For some 0 < α < 1, set It is easy to check that u satisfies Assumption 2.6, 3.4 and for any ζ > 0, In this section, we assume that the investors have the same recursive utility as above. In the following, we investigate three different kinds of wealth equations.

Linear wealth equation
In Example 2.1, suppose that r t ≡ 0 and σ t ≡ I d×d . Then, the wealth equation becomes where b t is a uniformly bounded progressively measurable process.
In this case,b ≡ 0 and Then the value functionṼ (ζ) in (3.2) becomes We conjecture thatṼ (t, γ, ζ) has the following form where (Ỹ ,Z) is the solution of the following BSDẼ The generator g(t, z) of (4.3) can be determined via the following martingale principle in [7]. The readers may also refer to Hu et al [13]. if and only ifṼ (t,γ, ζ) is a martingale.
Applying Itô's formula to According to Lemma 4.1, we have For z ∈ R d and t ∈ [0, T ], define By the result of Kobylanski [22], the quadratic BSDE (4.3) has a unique solution (Ỹ ,Z).

Then the infimum in (4.2) is attained at
The second infimum in (3.3) is attained at Thus, the optimal terminal wealth is given bŷ It is easy to check the following propositions.
Proposition 4.2 When b t is a deterministic function, we haveZ t = 0 and

Higher interest rate for borrowing
In this subsection, for simplicity, we assume that all variables are 1-dimentional.
Since b t is a deterministic function, g(t, z) is also deterministic. By the existence and uniqueness theorem of BSDE, the solution of (4.5) satisfiesZ t = 0, t ∈ [0, T ]. Thus the optimal solutionsμ andγ attain the infimum in It is easy to check that the following equations For the optimal solutionsμ andγ, we havẽ The second infimum in (3.3) is attained at Thus, the optimal terminal wealth is given bŷ We can easily deduce the following propositions.
This coincides with the result of Appendix B in Cvitanic and Karatzas [4].
, then the infimum in (4.6) attained atγ t = −K, and Remark 4.5 When b, R, r are bounded progressively measurable processes, g(t, z) is no longer deterministic. Similar analysis as in Theorem 7 of Hu et al [13], we can prove the quadratic BSDE (4.5) has a unique solution (Ỹ t ,Z t ). Thanks to the boundness, closeness and convexity of B 2 and B ′ 2 , there exists a pair (μ,γ) which attains the infimium of g(t,Z t ).
We haveṼ The second infimum in (3.3) is attained at Thus, the optimal terminal wealth is given bŷ It is easy to prove the following proposition.
Proposition 4.6 (δ,γ) may not be unique. But the optimal terminal wealth is unique. Furthermore, we have that for t ∈ [0, T ] and i = 1, ..., d, Now we employ the dynamic programming principle to calculate the optimal wealth process, the optimal portfolio strategiesas as well as the utility intensity process.
Suppose that the wealth of an large investor is x at time t. The wealth equation is where π ∈Ā(x; t, T ) (recall (2.9)). The recursive utility is v(t, x) satisfies the following HJB equation (refer to [25]): Let E γ [·] be the expectation operator with respect to P γ . Under P γ , the process is a Brownian motion.

s.t.
   π ∈Ā(x), (X t,x , π t,x ) satisfies the following equation (4.14), (4.13)    dX t,x s = (π ′ s b s − π ′ sδ s )ds + π ′ s dW s = (π ′ s b s + π ′ sγ s − π ′ sδ s )ds + π ′ s dWγ s X t,x t = x ≥ 0. (4.14) Therefore, v(t, x) also satisfies the following HJB equation: It is easy to verify the following theorem. By this theorem, we know the large investor will invest 1 1−α (b s +γ s −δ s ) percent of his wealth to the stocks. Especially, when |b it | ≤ K i + ε i , the investor will not invest on the ith stock at all.  where π + and π − denotes the d-dimensional vector with ith component π + i and π − i , i = 1, ..., d, respectively. This kind of wealth equation describes the different expected returns for long and short position of the stocks which is appeared in Jouini and Kallal [18] and El Karoui et al [10]. It also appeared in El Karoui et al [9] when there are taxes which must be paid on the gains made on the stocks. Remark 4.9 When b is a bounded progressively measurable processes, g(t, z) is no longer deterministic. We can prove the quadratic BSDE (4.8) has a unique solution (Ỹ t ,Z t ). There exists a pair (δ,γ) attains the infimium of g(t,Z t ) due to the boundness, closeness and convexity of B 2 and B ′ 2 .