# Recursive Utility Maximization for Terminal Wealth under Partial Information

Authors: Shaolin Ji; Xiaomin Shi
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

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 theK-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 perturbation method to characterize the optimal terminal wealth.


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## Body

## 1. 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 wealthX(T) on finite time interval [0,T], while the recursive utility is characterized by the initial value Y(0) of the following Backward stochastic differential equation (BSDE for short)(1)Yt=uXT+∫tTfs,Ys,Zsds-∫tTZsdW^s.The market consists of a riskless asset and d risky assets, the latter being driven by a d-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 G-framework. Hu et al. [8] introduced a BSDE driven by G-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 f in Cvitanić et al. [13] does not depend on z. 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 F0-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 generatorf 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 generatorf 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 Section2, 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 K-ignorance model and give explicit saddle points of several examples. In Section 5, we characterize the optimal wealth when the generator f is smooth.


## 2. The Problem of Recursive Utility Maximization under Partial Observation

### 2.1. Classical Formulation of the Problem

We consider a financial market consisting of a riskless asset whose price process is assumed for simplicity to be equal to one andd risky securities (the stocks) whose prices are stochastic processes Si(t), i=0,1,…,d governed by the following SDEs:(2)dSit=Sitμitdt+∑j=1dσijtdWjt,i=1,…,d,where W(·)=(W1·,…,Wd·)′ is a standard d-dimensional Brownian motion defined on a filtered complete probability space (Ω,F,{Ft}t≥0,P). μ′={μ′(t)=(μ1(t),…,μd(t)),t∈[0,T]} is the appreciation rate of the stocks which is Ft-adapted, bounded, and the d×d matrix σ(t)=(σij(t))1≤i,j≤d is the disperse rate of the stocks. Here and throughout the paper ′ denotes the transpose operator.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 byG={Gt}t≥0, which is the P-augmentation of the filtration generated by the price processes σ(S(u);0≤u≤t). The matrix disperse coefficient σ(t) is assumed invertible, bounded uniformly, and ∃ε>0, ρ′σ(t)σ′(t)ρ≥ερ2, ∀ρ∈Rd, t∈[0,T], a.s. In fact, σ(t) can be obtained from the quadratic variation of the price process. So we assume w.l.o.g. that σ(t) is Gt-adapted. However, the appreciation rate μ′(t)≔(μ1(t),…,μd(t)) is not observable for the investors.A small investor whose actions cannot affect the market prices can decide at timet∈[0,T] what amount πi(t) of his wealth to invest in the ith stock, i=1,…,d. Of course, his decision can only be based on the available information {Gt}t=0T; that is, the processes π′(·)=(π1(·),…,πd(·)):[0,T]×Ω→Rd are {Gt}t=0T progressively measurable and satisfy E∫0Tπ(t)2dt<∞.Then the wealth processX(·)≡Xx,π(·) of a self-financing investor who is endowed with initial wealth x>0 satisfies the following stochastic differential equation:(3)dXt=∑i=1dπitdSitSit=π′tμtdt+π′tσtdWt.Because the only information available to the investor isG, we could not use the Brownian motion W to define the recursive utility. As we will show in the following, there exists a Brownian motion W^ under P in the filtered measurable space (Ω,G) which is often referred to as an innovation process. The recursive utility process Y(t)≡Yx,π(t) of the investor is defined by the following backward stochastic differential equation:(4)Yt=uXT+∫tTfs,Ys,Zsds-∫tTZsdW^s,where f and u are functions satisfying the following assumptions.Assumption 1.
(A1)f:Ω×[0,T]×R×Rd→R is G-progressively measurable for any (y,z)∈R×Rd.
(A2) There exists a constantC≥0 such that(5)ft,y1,z1-ft,y2,z2≤Cy1-y2+z1-z2,∀t,ω,y1,y2,z1,z2∈Ω×0,T×R×R×Rd×Rd.(A3) f(t,·,·) is continuous about t and E∫0Tf2(t,0,0)dt<+∞.Assumption 2.
u : 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 (P,G)-Brownian motion W^.We introduce the following spaces:(6)L2Ω,GT,P;R≔ξ:Ω⟶R∣ξisGT-measurable,Eξ2<∞,MG20,T;Rd≔ϕ:0,T×Ω⟶Rd∣ϕt0≤t≤TisG-progressively  measurable  process,ϕ2=E∫0Tϕt2dt<∞,SG20,T;R≔ϕ:0,T×Ω⟶R∣ϕt0≤t≤TisG-progressively  measurable  process,ϕS2=Esup0≤t≤T⁡ϕt2<∞.For notational simplicity, we will often writeLGT2, MG2 and SG2 instead of L2(Ω,GT,P;R), MG2(0,T;Rd), and SG2(0,T;R), respectively.We will show in the next subsection that under Assumption1, for any ξ∈LGT2, the BSDE (4) has a unique solution (Y(·),Z(·))∈SG2×MG2. Then for each π∈MG2,X(T)∈LGT2, and Assumption 2 ensures that the variable u(X(T))∈LGT2. Thus, under Assumptions 1 and 2, the recursive utility process associated with this terminal value is well defined.Given an utility function satisfying Assumption2 and initial endowment x, the recursive utility maximization problem with bankruptcy prohibition is formulated as the investor chooses a portfolio strategy so as to(7)MaximizeYx,π0,s.t.Xt≥0,t∈0,T,a.s.,π·∈MG2,X·,π·satisfies  Eq.3,Y·,Z·satisfies  Eq.4,where X(t)≥0 means that no-bankruptcy is required.Definition 4.
A portfolioπ(·) is said to be admissible if π(·)∈MG2 and the corresponding wealth processes X(t)≥0, t∈[0,T], a.s.Given initial wealthx>0, denote by A¯(x) the set of investor’s feasible portfolio strategies; that is(8)A¯x=π:π∈MG2,Xx,πt≥0,dP⊗dta.s..


### 2.2. Reduction to a Problem under Full Information

Define the risk premium processη(t)=σ(t)-1μ(t). Because we have assumed the processes μ(·) and σ(·) are uniformly bounded, the process(9)Lt≔exp⁡-∫0tη′sdWs-12∫0tηs2dsis a (P,F) martingale. So a probability measure P~ can be defined by(10)P~A=ELTIA,∀A∈FT,wheredP~dP=LT.P~ is usually called risk neutral probability in the financial market. The process(11)W~t≔Wt+∫0tηsds,0≤t≤Tis a Brownian motion under P~ by Girsanov’s theorem.Then we can rewrite the stock price processes (2) as(12)dSit=Sit∑j=1dσijtdW~jt,i=1,…,d.Note thatσ(t) is assumed to be bounded, invertible, and Gt-adapted. So the filtration G coincides with the augmented natural filtration of W~ by Theorem V.3.7 in [19].Letη^(t)≔E[η(t)∣Gt] be a measurable version of the conditional expectation of η with respect to the filtration G. Set μ^: μ^(t)=E[μ(t)∣Gt]. Then μ^(t)=σ(t)η^(t), since σ is G-adapted.We introduce the process(13)W^t≔W~t-∫0tη^sds=Wt+∫0tηs-η^sds,t≥0.By Theorem8.1.3 and Remark 8.1.1 in Kallianpur [20], {W^(t),t≥0} is a (G,P)-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:(14)dSit=Sitμ^itdt+∑j=1dσijtdW^jt,i=1,…,d,dXt=π′tμ^tdt+π′tσtdW^t.Now all the coefficients in our model are observable. So we are in a full observation model and our problem (7) can be reformulated as follows:(15)MaximizeYx0,π0,s.t.Xt≥0∀t∈0,Ta.s.,π·∈MG2,X·,π·satisfies  Eq.3,Y·,Z·satisfies  Eq.4.


### 2.3. 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).Lemma 5.
Under Assumption1, for ∀ξ∈LG2, there exists a unique solution (Y,Z)∈SG2×MG2 to the BSDE (4).SinceG is strictly larger than the augmented filtration of the (P,G)-Brownian motion W^ in general, equation (4) is not a standard BSDE. Fortunately, by Theorem 8.3.1 in [20], every square integrable Gt-martingale M(t) can be represented as(16)Mt=M0+∫0tZ′sdW^s,where Z(·)∈MG2. Thus, applying similar analysis as in [21], it is easy to prove this lemma.Letq(·)≔σ(·)′π(·). Since σ(·) is invertible, q(·) can be regarded as the control variable instead of π(·). By the existence and uniqueness result of BSDE (4), selecting q(·) is equivalent to selecting the terminal wealth X(T). If we take the terminal wealth as control variable, the wealth equation and recursive utility process can be written as(17)-dXt=-q′-1tμ^tdt-q′tdW^t,XT=ξ,-dYt=ft,Yt,Ztdt-Z′tdW^t,YT=uξ,where the “control” is the terminal wealth ξ to be chosen from the following set:(18)U≔ξ∣ξ∈LGT2,ξ≥0.From now on, we denote the solution of (17) by (Xξ(·),qξ(·),Yξ(·),Zξ(·)). We also denote Xξ(0) and Yξ(0) by X0ξ and Y0ξ, respectively.As implied by the comparison theorem for BSDE (4), the nonnegative terminal wealth (ξ=X(T)≥0) keeps the wealth process nonnegative all the time. This gives rise to the following optimization problem:(19)MaximizeJξ≔Y0ξ,s.t.ξ∈U,X0ξ=x,Xξ·,qξ·,Yξ·,Zξ·satisfies  Eq.17.Definition 6.
A random variableξ∈U is called feasible for the initial wealth x if and only if Xξ(0)=x. We will denote the set of all feasible ξ for the initial wealth x by A(x).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 ξ 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 Xξ(0)=x that now becomes a constraint.Feasibleξ∗∈A(x) is called optimal if it attains the maximum of J(ξ) over A(x). Once ξ∗ is determined, the optimal portfolio can be obtained by solving the first equation in (17) with Xξ∗(T)=ξ∗.


## 2.1. Classical Formulation of the Problem

We consider a financial market consisting of a riskless asset whose price process is assumed for simplicity to be equal to one andd risky securities (the stocks) whose prices are stochastic processes Si(t), i=0,1,…,d governed by the following SDEs:(2)dSit=Sitμitdt+∑j=1dσijtdWjt,i=1,…,d,where W(·)=(W1·,…,Wd·)′ is a standard d-dimensional Brownian motion defined on a filtered complete probability space (Ω,F,{Ft}t≥0,P). μ′={μ′(t)=(μ1(t),…,μd(t)),t∈[0,T]} is the appreciation rate of the stocks which is Ft-adapted, bounded, and the d×d matrix σ(t)=(σij(t))1≤i,j≤d is the disperse rate of the stocks. Here and throughout the paper ′ denotes the transpose operator.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 byG={Gt}t≥0, which is the P-augmentation of the filtration generated by the price processes σ(S(u);0≤u≤t). The matrix disperse coefficient σ(t) is assumed invertible, bounded uniformly, and ∃ε>0, ρ′σ(t)σ′(t)ρ≥ερ2, ∀ρ∈Rd, t∈[0,T], a.s. In fact, σ(t) can be obtained from the quadratic variation of the price process. So we assume w.l.o.g. that σ(t) is Gt-adapted. However, the appreciation rate μ′(t)≔(μ1(t),…,μd(t)) is not observable for the investors.A small investor whose actions cannot affect the market prices can decide at timet∈[0,T] what amount πi(t) of his wealth to invest in the ith stock, i=1,…,d. Of course, his decision can only be based on the available information {Gt}t=0T; that is, the processes π′(·)=(π1(·),…,πd(·)):[0,T]×Ω→Rd are {Gt}t=0T progressively measurable and satisfy E∫0Tπ(t)2dt<∞.Then the wealth processX(·)≡Xx,π(·) of a self-financing investor who is endowed with initial wealth x>0 satisfies the following stochastic differential equation:(3)dXt=∑i=1dπitdSitSit=π′tμtdt+π′tσtdWt.Because the only information available to the investor isG, we could not use the Brownian motion W to define the recursive utility. As we will show in the following, there exists a Brownian motion W^ under P in the filtered measurable space (Ω,G) which is often referred to as an innovation process. The recursive utility process Y(t)≡Yx,π(t) of the investor is defined by the following backward stochastic differential equation:(4)Yt=uXT+∫tTfs,Ys,Zsds-∫tTZsdW^s,where f and u are functions satisfying the following assumptions.Assumption 1.
(A1)f:Ω×[0,T]×R×Rd→R is G-progressively measurable for any (y,z)∈R×Rd.
(A2) There exists a constantC≥0 such that(5)ft,y1,z1-ft,y2,z2≤Cy1-y2+z1-z2,∀t,ω,y1,y2,z1,z2∈Ω×0,T×R×R×Rd×Rd.(A3) f(t,·,·) is continuous about t and E∫0Tf2(t,0,0)dt<+∞.Assumption 2.
u : 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 (P,G)-Brownian motion W^.We introduce the following spaces:(6)L2Ω,GT,P;R≔ξ:Ω⟶R∣ξisGT-measurable,Eξ2<∞,MG20,T;Rd≔ϕ:0,T×Ω⟶Rd∣ϕt0≤t≤TisG-progressively  measurable  process,ϕ2=E∫0Tϕt2dt<∞,SG20,T;R≔ϕ:0,T×Ω⟶R∣ϕt0≤t≤TisG-progressively  measurable  process,ϕS2=Esup0≤t≤T⁡ϕt2<∞.For notational simplicity, we will often writeLGT2, MG2 and SG2 instead of L2(Ω,GT,P;R), MG2(0,T;Rd), and SG2(0,T;R), respectively.We will show in the next subsection that under Assumption1, for any ξ∈LGT2, the BSDE (4) has a unique solution (Y(·),Z(·))∈SG2×MG2. Then for each π∈MG2,X(T)∈LGT2, and Assumption 2 ensures that the variable u(X(T))∈LGT2. Thus, under Assumptions 1 and 2, the recursive utility process associated with this terminal value is well defined.Given an utility function satisfying Assumption2 and initial endowment x, the recursive utility maximization problem with bankruptcy prohibition is formulated as the investor chooses a portfolio strategy so as to(7)MaximizeYx,π0,s.t.Xt≥0,t∈0,T,a.s.,π·∈MG2,X·,π·satisfies  Eq.3,Y·,Z·satisfies  Eq.4,where X(t)≥0 means that no-bankruptcy is required.Definition 4.
A portfolioπ(·) is said to be admissible if π(·)∈MG2 and the corresponding wealth processes X(t)≥0, t∈[0,T], a.s.Given initial wealthx>0, denote by A¯(x) the set of investor’s feasible portfolio strategies; that is(8)A¯x=π:π∈MG2,Xx,πt≥0,dP⊗dta.s..


## 2.2. Reduction to a Problem under Full Information

Define the risk premium processη(t)=σ(t)-1μ(t). Because we have assumed the processes μ(·) and σ(·) are uniformly bounded, the process(9)Lt≔exp⁡-∫0tη′sdWs-12∫0tηs2dsis a (P,F) martingale. So a probability measure P~ can be defined by(10)P~A=ELTIA,∀A∈FT,wheredP~dP=LT.P~ is usually called risk neutral probability in the financial market. The process(11)W~t≔Wt+∫0tηsds,0≤t≤Tis a Brownian motion under P~ by Girsanov’s theorem.Then we can rewrite the stock price processes (2) as(12)dSit=Sit∑j=1dσijtdW~jt,i=1,…,d.Note thatσ(t) is assumed to be bounded, invertible, and Gt-adapted. So the filtration G coincides with the augmented natural filtration of W~ by Theorem V.3.7 in [19].Letη^(t)≔E[η(t)∣Gt] be a measurable version of the conditional expectation of η with respect to the filtration G. Set μ^: μ^(t)=E[μ(t)∣Gt]. Then μ^(t)=σ(t)η^(t), since σ is G-adapted.We introduce the process(13)W^t≔W~t-∫0tη^sds=Wt+∫0tηs-η^sds,t≥0.By Theorem8.1.3 and Remark 8.1.1 in Kallianpur [20], {W^(t),t≥0} is a (G,P)-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:(14)dSit=Sitμ^itdt+∑j=1dσijtdW^jt,i=1,…,d,dXt=π′tμ^tdt+π′tσtdW^t.Now all the coefficients in our model are observable. So we are in a full observation model and our problem (7) can be reformulated as follows:(15)MaximizeYx0,π0,s.t.Xt≥0∀t∈0,Ta.s.,π·∈MG2,X·,π·satisfies  Eq.3,Y·,Z·satisfies  Eq.4.


## 2.3. 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).Lemma 5.
Under Assumption1, for ∀ξ∈LG2, there exists a unique solution (Y,Z)∈SG2×MG2 to the BSDE (4).SinceG is strictly larger than the augmented filtration of the (P,G)-Brownian motion W^ in general, equation (4) is not a standard BSDE. Fortunately, by Theorem 8.3.1 in [20], every square integrable Gt-martingale M(t) can be represented as(16)Mt=M0+∫0tZ′sdW^s,where Z(·)∈MG2. Thus, applying similar analysis as in [21], it is easy to prove this lemma.Letq(·)≔σ(·)′π(·). Since σ(·) is invertible, q(·) can be regarded as the control variable instead of π(·). By the existence and uniqueness result of BSDE (4), selecting q(·) is equivalent to selecting the terminal wealth X(T). If we take the terminal wealth as control variable, the wealth equation and recursive utility process can be written as(17)-dXt=-q′-1tμ^tdt-q′tdW^t,XT=ξ,-dYt=ft,Yt,Ztdt-Z′tdW^t,YT=uξ,where the “control” is the terminal wealth ξ to be chosen from the following set:(18)U≔ξ∣ξ∈LGT2,ξ≥0.From now on, we denote the solution of (17) by (Xξ(·),qξ(·),Yξ(·),Zξ(·)). We also denote Xξ(0) and Yξ(0) by X0ξ and Y0ξ, respectively.As implied by the comparison theorem for BSDE (4), the nonnegative terminal wealth (ξ=X(T)≥0) keeps the wealth process nonnegative all the time. This gives rise to the following optimization problem:(19)MaximizeJξ≔Y0ξ,s.t.ξ∈U,X0ξ=x,Xξ·,qξ·,Yξ·,Zξ·satisfies  Eq.17.Definition 6.
A random variableξ∈U is called feasible for the initial wealth x if and only if Xξ(0)=x. We will denote the set of all feasible ξ for the initial wealth x by A(x).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 ξ 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 Xξ(0)=x that now becomes a constraint.Feasibleξ∗∈A(x) is called optimal if it attains the maximum of J(ξ) over A(x). Once ξ∗ is determined, the optimal portfolio can be obtained by solving the first equation in (17) with Xξ∗(T)=ξ∗.


## 3. Dual Method for Recursive Utility Maximization

In this section, we impose the following concavity condition.Assumption 7.
The function(y,z)↦f(ω,t,y,z) is concave for all (ω,t)∈Ω×[0,T].We also need the following assumption onu.Assumption 8.
u : ( 0 , ∞ ) → R is strictly increasing, strictly concave, and continuously differentiable, and satisfies(20)u′0+≔limx↓0⁡u′x=∞,u′∞≔limx→∞⁡u′x=0.Under Assumption8, Assumption 2 seems too restrictive and it precludes some interesting examples. So in the following two sections, for any given utility function u satisfying Assumption 8, we set(21)U=ξ∣ξ∈LGT2,uξ∈LGT2,ξ≥0.In this section, we assumeσ≡Id, the d-dimensional identity matrix. Let F(t,β,γ) be the Fenchel-Legendre transform of f:(22)Fω,t,β,γ≔supy,z∈R×Rd⁡fω,t,y,z-yβ-z′γ,β,γ∈R×Rd.Let the effective domain of F be DF≔{(ω,t,β,γ)∈Ω×[0,T]×R×Rd|F(ω,t,β,γ)<+∞}. As was shown in [22], the (ω,t)-section of DF, denoted by DF(ω,t) is included in the bounded domain B=[-C,C]d+1⊂R×Rd, where C is the Lipschitz constant of f.We have the duality relation by the concavity off,(23)fω,t,y,z=infβ,γ∈DFω,t⁡Fω,t,β,γ+yβ+z′γ.For every (ω,t,y,z) the infimum is achieved in this relation by a pair (β,γ) which depends on (ω,t).Set(24)B=β,γ∣β,γisG-progressively  measurable,B-valued,E∫0TFt,βt,γt2dt<+∞.Then B is a convex set. For any (β,γ)∈B, let(25)fβ,γt,y,z=Ft,βt,γt+yβt+z′γt,and denote by (Yβ,γ,Zβ,γ) the unique solution to the linear BSDE (4) with fβ,γ.By similar analysis as Proposition3.4 in [22], we have the following variational formulation of Xξ(t) and Yξ(t).Lemma 9.
Under Assumptions1 and 7, for any ξ∈U, the solutions (Xξ(·),qξ(·)),(Yξ(·),Zξ(·)) of (17) can be represented as (26)Xξt=L^-1tEL^Tξ∣Gt,Yξt=ess⁡infβ,γ∈B⁡Ytβ,γ,t∈0,T,a.s.,where (27)L^t≔e-∫0tμ^′sdW^s-1/2∫0tμ^s2ds,Ytβ,γ=E∫tTΓt,sβ,γFs,βs,γsds+Γt,Tβ,γuξ∣Gt,Γt,sβ,γ=e∫tsβr-1/2γr2dr+∫tsγr′dW^r.In particular, we have Yξ(0)=infβ,γ∈B⁡E[∫0TΓ0,sβ,γF(s,βs,γs)ds+Γ0,Tβ,γu(ξ)].Remark 10.
By Theorem3.1 in [11], we have L^(t)=E[L(t)|Gt], t∈[0,T], a.s.By Lemma9, A(x)={ξ∈U∣E[L^(T)ξ]=x}. Thus, our problem is equivalent to the following problem:(28)MaximizeJξ=infβ,γ∈B⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξs.t.ξ∈Ax.The maximum recursive utility that the investor can achieve is(29)V_x≔supξ∈Ax⁡infβ,γ∈B⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ.It is dominated by its “min-max” counterpart(30)V¯x≔infβ,γ∈B⁡supξ∈Ax⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ.If we can find(β^,γ^,ξ^)∈B×A(x) such that(31)V_x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^=V¯x,then the optimal solution of problem (28) is ξ^.Let us introduce the monotone decreasing functionI(·) as the inverse of the marginal utility function u′(·) and the convex dual(32)u~ζ≔maxx>0⁡ux-ζx=uIζ-ζIζ,ζ>0.Then,∀ξ∈A(x), ∀(β,γ)∈B, ∀ζ>0,(33)E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζξL^T=E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζx.Furthermore, we have equality in the above formula for someξ^∈A(x), (β^,γ^)∈B, ζ^>0 if and only if the conditions(34)Eξ^L^T=x,ξ^=Iζ^L^TΓ0,Tβ^,γ^,a.s.are satisfied simultaneously. And in this case, we have(35)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x.Lemma 11.
Under Assumptions1, 7, anfd 8, suppose that there exists a quadruple (ξ^,β^,γ^,ζ^)∈(A(x)×B×(0,∞)) which satisfies (34) and(36)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^,∀β,γ∈B.Then we have ∀ξ∈A(x), ∀(β,γ)∈B, and(37)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ≤E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^.That is, (ξ^,β^,γ^) is a saddle point satisfying (31).Proof.
We only prove the first relationship in (37). Let (β,γ)=(β^,γ^) and ζ=ζ^ in (33). We get (38)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ≤E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^,∀ξ∈Ax,by (35). This completes the proof.Let us introduce the value function(39)V~ζ≡V~ζ;x≔infβ,γ∈B⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ,0<ζ<∞.By (33), we have(40)V¯x≤V∗x,where(41)V∗x≔infζ>0,β,γ∈B⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζx=infζ>0⁡V~ζ+ζx.Lemma 12.
Under the assumptions of Lemma11, the followings hold:(i)
(β^,γ^) attains the infimum in (39) with ζ=ζ^.(ii)
The triple(ζ^,β^,γ^) attains the first infimum in (41).(iii)
The numberζ^∈(0,∞) attains the second infimum in (41).(iv)
There is no “duality gap” in (40); that is,(42)V∗x=V¯x=V_x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^.Proof.
(i) By (35) and (36), (43)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^-ζ^x≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^-ζ^x≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζ^L^TΓ0,Tβ,γ,∀β,γ∈B,where the last inequality is due to (33).
(ii) By (35) and (36), we have (44)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζx,∀β,γ∈B,∀ζ∈0,∞where the last inequality is an application of (33) to ξ=ξ^.
(iii) By (i), (35), and (36), (45)V~ζ^+ζ^x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζx,∀β,γ∈B,∀ζ∈0,∞.So we get V~(ζ^)+ζ^x≤infβ,γ∈B⁡E[∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~(ζ(L^T/Γ0,Tβ,γ))]+ζx=V~(ζ)+ζx, ∀ζ∈(0,∞).
(iv) By (ii) and (35), (46)V∗x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^=V¯x=V_x.This completes the proof.In the following, we prove the existence of the quadruple(ξ^,β^,γ^,ζ^) which is postulated in Lemma 11.Notice that the functionx↦xu~(1/x) is convex. By similar analysis as in Appendix B of [23], the following lemma holds.Lemma 13.
Under Assumptions1, 7, and 8, for any given ζ>0, there exists a pair (β^,γ^)=(β^ζ,γ^ζ)∈B which attains the infimum in (39).Lemma 14.
Under Assumptions1, 7, and 8, suppose(47)E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ<∞,∀ζ>0,∀β,γ∈B.Then for any given x>0, there exists a number ζ^=ζ^x∈(0,∞) which attains V∗(x)=infζ>0⁡[V~(ζ)+ζx].Proof.
⁡
Step 1. By the convexity of u~ and Lemma 13, V~(·) is convex. Fix ζ>0; denote (β^,γ^)=(β^ζ,γ^ζ) as in Lemma 13. For any δ>0, we have (48)V~ζ+δ-V~ζδ≤EΓ0,Tβ^,γ^u~ζ+δL^T/Γ0,Tβ^,γ^-Γ0,Tβ^,γ^u~ζL^T/Γ0,Tβ^,γ^δ≤EL^Tu~′ζ+δL^T/Γ0,Tβ^,γ^=-EL^TIζ+δL^T/Γ0,Tβ^,γ^.Then, by Levi’s lemma,(49)limδ→0+⁡V~ζ+δ-V~ζδ≤-EL^TIζL^TΓ0,Tβ^,γ^,limδ→0+⁡V~ζ-V~ζ-δδ≥-EL^TIζL^TΓ0,Tβ^,γ^.Since V~(·) is convex, we obtain that V~(·) is differentiable on (0,∞) and V~′(ζ)=-E[L^(T)I(ζ(L^T/Γ0,Tβ^,γ^))].
Step 2. Because μ(·) is bounded, we have that, for any ζ∈(0,∞), L^(T)/Γ0,Tβ^,γ^<+∞, a.s. Then,(50)V~′+∞≔limζ→+∞⁡V~′ζ=-limζ→∞⁡EL^TIζL^TΓ0,Tβ^,γ^=0,V~′0≔limζ→0+⁡V~′ζ=-limζ→0+⁡EL^TIζL^TΓ0,Tβ^,γ^=-∞.
Thus, there exists a numberζ^ which attains V∗(x) and V~′(ζ^)=-x∈(-∞,0). This completes the proof.Lemma 15.
Under Assumptions1, 7, and 8, V∗(x)=E[∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u(ζ^(L^T/Γ0,Tβ^,γ^))]+ζ^x with ζ^=ζ^x as in Lemma 14 and (β^,γ^)=(β^ζ^,γ^ζ^) as in Lemma 13.Proof.
We have(51)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^+ζ^x=V~ζ^+ζ^x≤V~ζ+ζx≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζL^TΓ0,Tβ,γ+ζx,∀β,γ∈B,∀ζ∈0,∞,∀x>0.This completes the proof.Our main result is the following theorem.Theorem 16.
Under Assumptions1, 7, and 8, let (ζ^,β^,γ^) as in Lemma 15 and define ξ^=I(ζ^(L^T/Γ0,Tβ^,γ^))a.s. If ξ^∈U, then (ζ^,β^,γ^,ξ^) satisfies all the conditions in Lemma 11, that is, (34) and (36).Proof.
Notice that(52)V~ζ^=infβ,γ∈B⁡E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γu~ζ^L^TΓ0,Tβ,γ=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^.Applying the maximum principle in Peng [24], we obtain a necessary condition for (β^,γ^):(53)Ft,βt,γt+ptβt+qtγt≥Ft,β^t,γ^t+ptβ^t+qtγ^t,∀β,γ∈B,where (pt,qt) is the solution of the adjoint equation(54)-dpt=Ft,β^t,γ^t+ptβ^t+qt′γ^tdt-qt′dW^t,pT=uIζ^L^TΓ0,Tβ^,γ^.
∀ ( β , γ ) ∈ B; let (yt,zt) and (y~t,z~t) be the unique solutions of the following two linear BSDEs:(55)yt=uξ^+∫tTysβ^s+zs′γ^s+Fs,β^s,γ^sds-∫tTzs′dW^s,(56)y~t=uξ^+∫tTy~sβs+z~s′γs+Fs,βs,γsds-∫tTz~s′dW^s.
By (53) and the comparison theorem of BSDE, we have yt≤y~t, t∈[0,T], a.s., especially y0≤y~0.
Solving the above linear BSDEs gives(57)y0=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^,y~0=E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^.
So(58)E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^uξ^≤E∫0TΓ0,sβ,γFs,βs,γsds+Γ0,Tβ,γuξ^,∀β,γ∈B,which exactly is (36).
By Lemma14, V~′(ζ^)=-x. By Lemma 13,(59)V~ζ^=E∫0TΓ0,sβ^,γ^Fs,β^s,γ^sds+Γ0,Tβ^,γ^u~ζ^L^TΓ0,Tβ^,γ^.Differentiating both sides of (59) as functions of ζ^, we get(60)EIζ^L^TΓ0,Tβ^,γ^L^T=x.This completes the proof.Remark 17.
It is worth pointing out that the adjoint processpt in the proof of the above theorem coincides with the optimal utility process yt in (55).


## 4.K-Ignorance

In this section, we study a special case which is calledK-ignorance by Chen and Epstein [25]. In this case, the generator f is specified as(61)ft,y,z=-Kz,K≥0.Chen and Epstein interpreted the term Kz as modeling ambiguity aversion rather than risk aversion. f(z)=-Kz is not differentiable. But it is concave and f(z)=infγ≤K⁡(γz). Then our results in the above section are still applicable.In this section, we assumed=1, σ≡1. The wealth equation and recursive utility become(62)-dXt=-q′tμ^tdt-q′tdW^t,XT=ξ,-dYt=-KZtdt-Z′tdW^t,YT=uξ.Our problem is formulated as follows:(63)MaximizeJξ≔Y0ξ,s.t.ξ∈U,X0=x,X·,q·,Y·,Z·satisfies  Eq.62.Now Lemma9 can be simplified to the following lemma.Lemma 18.
Forξ∈U, the solutions (X(·),q(·)) and (Y(·),Z(·)) of (62) can be represented as (64)Xt=L^-1tEL^Tξ∣Gt,Yt=ess⁡infγ∈B⁡Γ0,t0,γ-1tEΓ0,T0,γuξ∣Gt,where (65)L^t=e-∫0tμ^sdW^s-1/2∫0tμ^s2ds,Γ0,t0,γ=e∫0tγsdW^s-1/2∫0tγs2ds,B=γ=γtt≥0∣γtisG-progressivelymeasurable,γt≤K,t∈0,T,a.s..For anyγ∈B, Γ0,t0,γ is (G,P)-martingale. Then, a new probability measure Pγ is defined on GT by(66)dPγdP=Γ0,T0,γand W^γ(t)=W^(t)-∫0tγsds is a Brownian motion under Pγ. Thus, Y(0)=infγ∈B⁡Eγ[u(ξ)] where Eγ[·] is the expectation operator with respect to Pγ.Our problem (63) is equivalent to the following problem:(67)MaximizeJξ=infγ∈B⁡Eγuξs.t.ξ∈Ax.The auxiliary dual problem in (39) becomes(68)V~ζ≡V~ζ;x≔infγ∈B⁡Eγu~ζZγT,0<ζ<∞,where Zγ(t)≔L^(t)/Γ0,t0,γ, t∈[0,T] a.s. and(69)V∗x≔infζ>0,γ∈B⁡Eγu~ζZγT+ζx=infζ>0⁡V~ζ+ζx.Applying the procedure in the previous section, we can find the saddle point. So we list the results without proof except Lemma19 in which a new proof is given.Lemma 19.
Under Assumption8, for any given ζ>0, there exists a unique γ^=γ^ζ∈B which attains the infimum in (68).Proof.
SetB′={Γ0,T0,γ∣γ∈B}, M={Mγ(T)≔Γ0,T0,γ/L^(T)∣γ∈B}, and g(x)=xu~(ζ/x) for x>0. Then problem (68) becomes(70)V~ζ=infMγT∈M⁡E~MγTu~ζ1MγT=infMγT∈M⁡E~gMγT,where E~[·] is the expectation operator with respect to the risk neutral measure P~. By Theorem 2.1 in [25], we know B′ is norm closed in L1(Ω). So B is closed under a.s. convergence because B is uniformly integrable. As a consequence, M is closed under a.s. convergence.
Consider a minimizing sequence{Mγn(T)}n≥1 for (70); that is(71)limn→∞⁡E~gMγnT=V~ζ.By Komlos’ theorem, there exists a sequence M¯γn(T)∈ conv(Mγn(T),Mγn+1(T),…,); that is, M¯γn(T)=∑k=nTnλkMγk(T), λk∈[0,1], and ∑k=nTnλk=1, such that the sequence {Mγn(T)}n≥1 converges a.s. to a random variable M. By a.s. closedness of M, we have M∈M; that is, ∃γ^∈B, s.t. M=Mγ^(T). Note that g is a strictly convex continuous function; we have (72)E~gM=E~limn→∞⁡gM¯γnT≤liminfn→∞⁡E~gM¯γnT≤liminfn→∞⁡λk∑k=nTnE~gMγkT=liminfn→∞⁡E~gMγnT=V~ζ.The uniqueness follows from the strict convexity of g. This completes the proof.Lemma 20.
Under Assumption8, if E~[I(ζZγ(T))]<∞, ∀ζ>0, ∀γ∈B, then for any given x>0, there exists a number ζ^=ζ^x∈(0,∞) which attains the infimum of V∗(x)=infζ>0⁡[V~(ζ)+ζx].Lemma 21.
Under Assumption8, V∗(x)=Eγ^u~(ζ^Zγ^(T))+ζ^x with ζ^=ζ^x as in Lemma 20 and γ^=γ^ζ^ as in Lemma 19.Theorem 22.
Under Assumption8, let (ζ^,γ^) be the same as in Lemma 21; then the optimal terminal wealth of problem (67) is(73)ξ^=Iζ^Zγ^T,a.s.if ξ^ belongs to U.In the following, we give some examples to illustrate our above analysis.Example 23 (constant absolute risk aversion).
Suppose thatu(x)=1-e-αx, x∈R, α>0, and the wealth of the investor may be negative. This utility function u does not satisfy Assumption 8. But it satisfies the following assumption.Assumption 24.
u is strictly increasing, strictly concave, and continuously differentiable, and(74)u′-∞≔limx↓-∞⁡u′x=∞,u′∞≔limx→∞⁡u′x=0.
Note that under Assumption24, the wealth process of the investor is allowed to be negative; the results in this section still hold.
For this example,I(ζ)=-1/αln⁡(ζ/α), ζ>0, and u~(ζ)=1-ζ/α+ζ/αln⁡(ζ/α), ζ>0. Then the value function of the auxiliary dual problem (68) is (75)Eγu~ζZγT=1-ζα+ζαln⁡ζα+ζαE~ln⁡ZγT=1-ζα+ζαln⁡ζα+ζ2αE~∫0Tμ^t+γt2dt,ζ>0.Apparently, γ^t (optimal γt) which attains the infimum of Problem (68) is independent of ζ. It is easy to see that(76)γ^t=-K∨-μ^t∧K,t∈0,T,a.s.The optimal value of problem (68) is(77)V~ζ=1-ζα+ζαln⁡ζα+ζ2αE~∫0Tμ^t+γ^t2dt,and the Lagrange multiplier in Lemma 20 is(78)ζ^≡ζ^x=αe-1/2E~∫0Tμ^t+γ^t2dt-αx=arg⁡minζ>0⁡V~ζ+ζx.
Thus, the optimal terminal wealth in Theorem22 is(79)ξ^=-1αln⁡ζ^Zγ^Tα.
Moreover, it is easy to check that(Y(t),Z(t))≔(1-(ζ^/α)Zγ^(t),(ζ^/α)(μ^(t)+γ^t)Zγ^(t)) and t∈[0,T] uniquely solves the utility equation in (62) when ξ=ξ^.
Note thatμ^(t) and γ^t are bounded; it is easy to verify ξ^∈LGT2 and u(ξ^)∈LGT2.Example 25 (logarithmic utility function).
Supposeu(x)=ln⁡x, x>0. In this case,(80)Iζ=1ζ,u~ζ=-ln⁡ζ-1,ζ>0.Then the value function of the auxiliary dual problem (68) is (81)Eγu~ζZγT=Eγ-ln⁡ζZγT-1=Eγ-ln⁡ZγT-ln⁡ζ-1=12Eγ∫0Tμ^t+γt2dt-ln⁡ζ-1,ζ>0.So the optimal γ^t is independent of ζ. Consider the following BSDE:(82)yγt=Eγ∫tTμ^s+γs2ds∣Gt=∫tTμ^s+γs2+γszγsds-∫tTzγsdW^s.
Set(83)ft,zt=infγ∈B⁡μ^t+γt2+γtzt=K2-2Kμ^t-Kzt+μ^t2,if-2μ^t+2K<zt;-14zt2-μ^tzt,if-2μ^t-2K≤zt≤-2μ^t+2K;K2+2Kμ^t+Kzt+μ^t2,ifzt<-2μ^t-2K,t∈0,T,a.s.It is easy to show that f(t,zt) is uniformly Lipschitz, so the following BSDE has a unique solution which we still denoted by (yt,zt).(84)yt=∫tTfs,zsds-∫tTzsdW^s.Then the infimum in problem (68) is attained at (85)γ^t=arg⁡infγ∈B⁡μ^t+γt2+γtzt=-KI-2μ^t+2K<zt+-μ^t-zt2I-2μ^t-2K≤zt≤-2μ^t+2K+KIzt<-2μ^t-2K,t∈0,T,a.s.
The Lagrange multiplier in Lemma20 is(86)ζ^≡ζ^x=1x=arg⁡minζ>0⁡V~ζ+ζx.
The optimal terminal wealth in Theorem22 is(87)ξ^=xZγ^T.
Note thatμ^(t) and γ^t are bounded; it is easy to verify ξ^∈LGT2 and u(ξ^)∈LGT2.Example 26.
Suppose that the appreciation rateμ(t) is a bounded deterministic function of t. In this case, Gt=Ft, t≥0, and we claim that(88)γ^t=-K∨-μt∧K,t∈0,T.Proof.
We show thatγ^ defined above attains the infimum of (68). Denote(89)vt,x≡vt,x;ζ≔E~gxMγ^TMγ^t.Then v(t,x) is the solution of the partial differential equation ∂v/∂t+(1/2)(∂2v/∂x2)x2(μt+γ^t)2=0.
∀ γ ∈ B; applying Itô’s formula to v(t,Mγ(t)), we have (90)dvt,Mγt=∂v∂t+12∂2v∂x2Mγt2μt+γt2dt+∂v∂xMγtμt+γtdW~t=12∂2v∂x2Mγt2μt+γt2-μt+γ^t2dt+∂v∂xMγtμt+γtdW~t.By the definition of γ^t (88), we have (μ(t)+γt)2-(μ(t)+γ^t)2≥0, t∈[0,T]. The convexity of v(t,·) guarantees that v(t,Mγ(t)) is a submartingale. Thus, ∀γ∈B,(91)Eγu~ζZγT=E~gMγT=E~vT,MγT≥E~v0,Mγ0=E~gMγ^T=Eγ^u~ζZγ^T.This completes the proof.Example 27.
Suppose thatμ·≤K, a.e., a.s. Then we have (92)γ^t=-μ^t,t∈0,T,a.s.Note that μ belongs to B when μ·≤K, a.e. a.s. Due to the convexity of g, we have that ∀γ∈B,(93)E~gMγt≥gE~MγT=g1≡gMγ^T≡E~gMγ^T.In this case, Pγ^ coincides with the risk neutral probability P~ on GT which leads to the optimal terminal wealth ξ^=x. This means that the investor will not invest in the risk assets at all.Remark 28.
For recursive utility for consumption of the form(94)Yt=uXT+∫tTucs+fs,Ys,Zsds-∫tTZsdW^s,where u and f satisfy Assumptions 2 and 8 and Assumptions 1 and 7, respectively, our method still works well.


## 5. Terminal Perturbation Method

When the generator of the recursive utility (4) is nonconcave, the dual method is not applicable. In this case, we apply the terminal perturbation method to obtain a characterization of the optimal terminal wealth. We need the following smooth assumption.Assumption 29.
f is continuously differentiable in (y,z).Letξ∗ be an optimal terminal wealth for (19); that is,(95)Yξ∗0=supξ∈Ax⁡Yξ0,And let (X∗(·),q∗(·),Y∗(·),Z∗(·)) be the corresponding state processes of (17).Set(96)Ω¯≔ω∈Ω∣ξ∗ω=0.By the terminal perturbation method in [17, 18], we have the following stochastic maximum principle.Theorem 30.
Under Assumptions1, 2, and 29, if ξ∗ is the optimal wealth of problem (19), then there exist h0∈R, h1≥0, and h02+h12=1 such that(97)h0mT+h1u′ξ∗nT≥0,a.s.onΩ¯;h0mT+h1u′ξ∗nT=0,a.s.onΩ¯c,where(98)dmt=-μ^′tσ′-1tmtdW^t,m0=1;dnt=fY∗tntdt+fZ∗′tntdW^t,n0=1,and fY∗(t)≔fY(t,Y∗(t),Z∗(t)), fZ∗(t)≔fZ(t,Y∗(t),Z∗(t)).Remark 31.
Note that we do not need the concavity property ofu in the above theorem.



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*Source: 2813707-2016-12-14.xml*