# Relationship between Maximum Principle and Dynamic

Authors: Jingtao Shi; Zhiyong Yu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the relationship between maximum principle and
dynamic programming for stochastic recursive optimal control problems. Under certain differentiability conditions, relations among the adjoint processes, the generalized Hamiltonian
function, and the value function are given. A linear quadratic recursive utility portfolio optimization problem in the financial engineering is discussed as an explicitly illustrated example of
the main result.


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## Body

## 1. Introduction

The nonlinearbackward stochastic differential equation (BSDE) was introduced by Pardoux and Peng [1]. Independently, Duffie and Epstein [2] introduced BSDE from economic background. In [2], they presented a stochastic differential formulation of recursive utility. Recursive utility is an extension of the standard additive utility with the instantaneous utility depending not only on the instantaneous consumption rate but also on the future utility. As found by El Karoui et al. [3], the utility process can be regarded as a solution to a special BSDE. The optimal control problem that the cost functional is described by the solution to a BSDE is called a stochastic recursive optimal control problem. In this case, the control systems become forward-backward stochastic differential equations (FBSDEs). This kind of optimal control problems has found important applications in real-world problems such as mathematical economics, mathematical finance, and engineering (see Schroder and Skiadas [4], El Karoui et al. [3, 5], Ji and Zhou [6], Williams [7], and Wang and Wu [8]).It is well known that Pontryagin’s maximum principle and Bellman’s dynamic programming are two of the most important tools in solving stochastic optimal control problems. See the famous reference book by Yong and Zhou [9] for systematic discussion. For stochastic recursive optimal control problems, Peng [10] first obtained a maximum principle when the control domain is convex. And then Xu [11] studied the nonconvex control domain case, but he needs to assume that the diffusion coefficient does not contain the control variable. Ji and Zhou [6] established a maximum principle when the forward state is constrained in a convex set at the terminal time. Wu [12] established a general maximum principle, where the control domain is nonconvex and the diffusion coefficient depends on the control variable. Maximum principle for stochastic recursive optimal control systems with Poisson jumps, and their applications in finance were studied in Shi and Wu [13], where the control domain is convex.For another important approach—dynamic programming—to study stochastic recursive optimal control problems, Peng [14] (also see Peng [15]) first obtained the generalized dynamic programming principle and introduced a generalized Hamilton-Jacobi-Bellman (HJB) equation which is a second-order parabolic partial differential equation (PDE). Result that the value function is the viscosity solution to the generalized HJB equation is also proved in [14]. Wu and Yu [16] extended the results of [14, 15] with obstacle constraint for the cost functional described by the solution to a reflected backward stochastic differential equation and proved that the value function is the unique viscosity solution to their generalized HJB equation. Li and Peng [17] generalized the results of [14, 15] by considering the cost functional defined by the controlled BSDE with jumps. They proved that the value function was the viscosity solution to the associated generalized HJB equation with integral-differential operators.Hence, a natural question arises: are there any relations between these two methods? Such a topic was intuitively discussed by Bismut [18] and Bensoussan [19] and then studied by many researchers. Under certain differentiability conditions, the relationship between the maximum principle and dynamic programming is essentially the relationship between the derivatives of the value function and the solution to the adjoint equation along the optimal state. However, the smoothness conditions do not hold in general and are difficult to verify a priori, see Zhou [20] for the deterministic case and Yong and Zhou [9] for its stochastic counterpart. Zhou [21] first obtained the relationship between general maximum principle and dynamic programming using the viscosity solution theory (see also Zhou [22] or Yong and Zhou [9]), without the assumption that the value function is smooth. For diffusion with jumps, the relationship between maximum principle and dynamic programming was first given by Framstad et al. [23, 24] under certain differentiability conditions, and then Shi and Wu [25] eliminated these restrictions within the framework of viscosity solutions. For singular stochastic optimal control problem, the relationship between maximum principle and dynamic programming was given by Bahlali et al. [26], with the derivatives of the value function. For Markovian regime-switching jump diffusion model, the relationship between maximum principle and dynamic programming was given by Zhang et al. [27], also with the derivatives of the value function.In this paper, we derive the relationship between maximum principle and dynamic programming for the stochastic recursive optimal control problem. For this problem, we connect the maximum principle of [10] with the dynamic programming of [14, 15], under certain differentiability conditions. Specifically, when the value function is smooth, we give relations among the adjoint processes, the generalized Hamiltonian function, and the value function. For this target, in Section 2, we first adopt some related results of [14, 15], which in this paper are stated as a stochastic verification theorem. Also we prove that under additional convexity conditions, the necessary conditions in the maximum principle of [10] are in fact sufficient. In Section 3, we show the relationship between maximum principle and dynamic programming under certain differentiability conditions for our stochastic recursive optimal control problem, by the martingale representation technique. In Section 4, we discuss a linear quadratic (LQ) recursive utility portfolio optimization problem in the financial engineering. In this problem, the state feedback optimal control is obtained by both the maximum principle and dynamic programming approaches, and the relations we obtained are illustrated explicitly. Finally, we end this paper with some concluding remarks in Section 5.Notations. Throughout this paper, we denote by Rn the space of n-dimensional Euclidean space, by Rn×d the space of n×d matrices, and by 𝒮n the space of n×n symmetric matrices. 〈·,·〉 and |·| denote the scalar product and norm in the Euclidean space, respectively. ⊤ appearing in the superscripts denotes the transpose of a matrix.


## 2. Problem Statement and Preliminaries

Let(Ω,ℱ,P) be a complete probability space equipped with a d-dimensional standard Brownian motion {W(t)}t≥0. For fixed t≥0, the filtration {ℱst}s≥t is generated as ℱst=σ{W(r)-W(t);t≤r≤s}⋁𝒩, where 𝒩 contains all P-null sets in ℱ and σ1⋁σ2 denotes the σ-field generated by σ1∪σ2. In particular, if t=0, we write ℱs≡ℱst.LetT>0 be finite and let U⊂Rk be nonempty, convex. For any initial time and state (t,x)∈[0,T)×Rn, consider the state Xt,x;u(·)∈Rn given by the following controlled SDE:
(1)dXt,x;u(s)=b(s,Xt,x;u(s),u(s))ds+σ(s,Xt,x;u(s),u(s))dW(s),s∈[t,T],Xt,x;u(t)=x.
Here b:[0,T]×Rn×U→Rn, σ:[0,T]×Rn×U→Rn×d are given functions.Givent∈[0,T), we denote by 𝒰[t,T] the set of {ℱst}s≥t-adapted processes. For given u(·)∈𝒰[t,T] and x∈Rn, an Rn-valued process Xt,x;u(·) is called a solution to (1) if it is an ℱst-adapted process such that (1) holds. We refer to such u(·)∈𝒰[t,T] as an admissible control and (Xt,x;u(·),u(·)) as an admissible pair. We assume the following.(H1)
b, σ are uniformly continuous in (s,x,u), and there exists a constant C>0 such that for all s∈[0,T], x,x^∈Rn, u,u^∈U,
(2)|b(s,x,u)-b(s,x^,u^)|+|σ(t,x,u)-σ(s,x^,u^)|≤C(|x-x^|+|u-u^|),|b(s,x,u)|+|σ(s,x,u)|≤C(1+|x|).For any u(·)∈𝒰[t,T], under (H1), SDE (1) has a unique solution Xt,x;u(·) by the classical SDE theory (see, e.g., Yong and Zhou [9]).Next, we introduce the following controlled BSDE coupled with controlled SDE (1):
(3)-dYt,x;u(s)=f(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u(s))ds-Zt,x;u(s)dW(s),s∈[t,T],Yt,x;u(T)=ϕ(Xt,x;u(T)).
Here f:[0,T]×Rn×R×Rd×U→R, Φ:Rn→R are given functions. We assume the following.(H2)
f, ϕ are uniformly continuous in (s,x,y,z,u) and there exists a constant C>0 such that for all s∈[0,T], x,x^∈Rn, y,y^∈R, z,z^∈Rd, u,u^∈U,
(4)|f(s,x,y,z,u)-f(s,x^,y^,z^,u^)|≤C(|x-x^|+|y-y^|+|z-z^|+|u-u^|),|f(s,x,0,0,u)|+|ϕ(x)|≤C(1+|x|),|ϕ(x)-ϕ(x^)|≤C|x-x^|.Then for any u(·)∈𝒰[t,T] and the given unique solution Xt,x;u(·) to (1), under (H2), BSDE (3) admits a unique solution (Yt,x;u(·),Zt,x;u(·)) by the classical BSDE theory (see Pardoux and Peng [1] or Yong and Zhou [9]).Givenu(·)∈𝒰[t,T], we introduce the cost functional
(5)J(t,x;u(·)):=-Yt,x;u(s)|s=t,(t,x)∈[0,T]×Rn.Our recursive stochastic optimal control problem is the following.Problem 1 (RSOCP).
For given(t,x)∈[0,T)×Rn, to minimize (5) subject to (1)–(3) over 𝒰[t,T].
We define the value function(6)V(t,x):=infu(·)∈𝒰[t,T]J(t,x;u(·)),(t,x)∈[0,T]×Rn,V(T,x)=-ϕ(x),x∈Rn.
Any u-(·)∈𝒰[t,T] achieves the above infimum is called an optimal control, and the corresponding solutions X-t,x;u-(·) to (1) and (Y-t,x;u-(·),Z-t,x;u-(·)) to (3) are called optimal state. For simplicity, we refer to (X-t,x;u-(·),Y-t,x;u-(·),Z-t,x;u-(·),u-(·)) as the optimal quartet.Remark 1.
Becauseb, σ, f, g are all deterministic functions, then from [15, Proposition 5.1 of Peng], we know that under (H1) and (H2), the above value function is a deterministic function. So our definition (6) is meaningful.
We introduce the following generalized HJB equation:(7)-vs(t,x)+supu∈UG(t,x,-v(t,x),-vx(t,x),-vxx(t,x),u)=0,(t,x)∈[0,T)×Rn,v(T,x)=-ϕ(x),∀x∈Rn,
where the generalized Hamiltonian function G:[0,T]×Rn×R×Rn×𝒮n×U→R is defined as
(8)G(t,x,r,p,A,u)≔12tr{σ(t,x,u)⊤Aσ(t,x,u)}+〈p,b(t,x,u)〉+f(t,x,r,σ(t,x,u)⊤p,u).
We have the following result.Lemma 2 (stochastic verification theorem).
Let (H1)-(H2) hold and let(t,x)∈[0,T)×Rn be fixed. Suppose that V∈C1,2([0,T]×Rn) is a solution to (7), then
(9)V(t,x)≤J(t,x;u(·)),∀u(·)∈𝒰[t,T],(t,x)∈[0,T]×R.
Furthermore, an admissible pair (X-t,x;u-(·),u-(·)) is optimal for Problem (RSOCP) if and only if
(10)G(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),-Vx(s,X-t,x;u-(s)),aaa-Vxx(s,X-t,x;u-(s)),u-(s))=maxu∈UG(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaa-Vx(s,X-t,x;u-(s)),-Vxx(s,X-t,x;u-(s)),u),
a.e. s∈[t,T], P-a.s.Proof.
For anyu(·)∈𝒰[t,T] with the corresponding state Xt,x;u(·), applying Itô’s formula to V(s,Xt,x;u(s)), we obtain the following:
(11)V(t,x)=-𝔼ϕ(Xt,x;u(T))-𝔼∫tT‍{(s,Xt,x;u(s),u(s))⊤Vs(s,X-t,x;u-(s))aaaaaaaaaa+〈Vx(s,Xt,x;u(s)),b(s,Xt,x;u(s),u(s))〉aaaaaaaaaa+12tr(σ(s,Xt,x;u(s),u(s))⊤aaaaaaaaaaaaaaa×Vxx(s,Xt,x;u(s))aaaaaaaaaaaaaaa×(s,Xt,x;u(s),u(s))⊤σ(s,Xt,x;u(s),u(s)))}ds=-𝔼ϕ(Xt,x;u(T))-𝔼∫tT‍{(s,Xt,x;u(s),u(s))⊤Vs(s,Xt,x;u(s))aaaaaaaaaaaaaa+G(s,Xt,x;u(s),-V(s,Xt,x;u(s)),aaaaaaaaaaaaaaaaaaa-Vx(s,Xt,x;u(s)),aaaaaaaaaaaaaaaaaaa-Vxx(s,Xt,x;u(s)),u(s))aaaaaaaaaaaaaa-f((s,Xt,x;u(s),u(s))⊤s,Xt,x;u(s),-V(s,Xt,x;u(s)),aaaaaaaaaaaaaaaaaaa-σ(s,Xt,x;u(s),u(s))⊤aaaaaaaaaaaaaaaaa×Vx(s,Xt,x;u(s)),u(s)(s,Xt,x;u(s),u(s))⊤)}ds=J(t,x;u(·))+𝔼∫tT‍{(s,Xt,x;u(s),-V(s,Xt,x;u(s)),-Vx(s,Xt,x;u(s)),-Vxx(s,Xt,x;u(s)),u(s))-Vs(s,Xst,x;u)aaaaaaaaa+G(s,Xt,x;u(s),-V(s,Xt,x;u(s)),aaaaaaaaa-Vx(s,Xt,x;u(s)),aaaaaaaaa-Vxx(s,Xt,x;u(s)),u(s)Xt,x;u(s))Vs(s,Xst,x;u)}ds≤J(t,x;u(·))+𝔼∫tT‍{-Vs(s,Xt,x;u(s))aaaaaaaaaaaaaa+maxu∈UG(s,Xt,x;u(s),-V(s,Xt,x;u(s)),aaaaaaaaaaaaaaaaaaaaaaa-Vx(s,Xt,x;u(s)),aaaaaaaaaaaaaaaaaaaaaaa-Vxx(s,Xt,x;u(s)),u)}ds=J(t,x;u(·)).
Thus (9) holds. Next, applying the above inequality to (X-t,x;u-(·),u-(·)), we have
(12)V(t,x)=J(t,x;u-(·))+𝔼∫tT‍{-Vs(s,X-t,x;u-(s))aaaaaaaaaaa+G(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaaaaaa-Vx(s,X-t,x;u-(s)),aaaaaaaaaaaaaa-Vxx(s,X-t,x;u-(s)),u-(s))}ds.
The desired result follows immediately from the fact that
(13)-Vs(s,X-t,x;u-(s))+G(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),-Vx(s,X-t,x;u-(s)),-Vxx(s,X-t,x;u-(s)),u-(s))≤0,
which is due to the generalized HJB equation (7). The proof is complete.In convenient to state the maximum principle, we regard the above controlled SDE (1) and BSDE (3) as a controlled FBSDE:
(14)dXt,x;u(s)=b(s,Xt,x;u(s),u(s))ds+σ(s,Xt,x;u(s),u(s))dW(s),-dYt,x;u(s)=f(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u(s))ds-Zt,x;u(s)dW(s),s∈[t,T],Xt,x;u(t)=x,Yt,x;u(T)=ϕ(Xt,x;u(T)).This kind of control system was studied by Peng [10] and a maximum principle was obtained. In order to mention his result, we need the following assumption.(H3)
b, σ, ϕ are continuously differentiable in (x,u) and f is continuously differentiable in (x,y,z,u). Moreover, bx, σx, fx, fy, fz, bu, σu, fu are bounded and there exists a constant C>0 such that
(15)|ϕx(x)|≤C(1+|x|),∀x∈Rn.Let(X-t,x;u-(·),Y-t,x;u-(·),Z-t,x;u-(·),u-(·)) be an optimal quartet. For all s∈[0,T], we denote
(16)b-(s):=b(s,X-t,x;u-(s),u-(s)),σ-(s):=b(s,X-t,x;u-(s),u-(s)),f-(s):=f(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),u-(s)),
and similar notations are used for all their derivatives.We introduce the adjoint equation which is an FBSDE:(17)-dp(s)=[b-x(s)⊤p(s)-f-x(s)⊤q(s)+σ-x(s)k(s)]ds-k(s)dW(s),dq(s)=f-y(s)⊤q(s)ds+f-z(s)⊤q(s)dW(s),s∈[t,T],p(T)=-ϕx(X-t,x;u-(T))⊤q(T),q(t)=1,
and the Hamiltonian function H:[0,T]×Rn×R×Rd×U×Rn×R×Rn×d→R is defined as
(18)H(t,x,y,z,u,p,q,k)≔〈p,b(t,x,u)〉-〈q,f(t,x,y,z,u)〉+tr[σ(t,x,u)⊤k].Under (H1)–(H3), the forward equation in (17) admits an obvious unique solution p(·), and then the backward equation in (17) admits a unique solution (q(·),k(·)). We call p, q, k the adjoint processes. Next, the following result holds.Lemma 3 (necessary maximum principle).
Let (H1)–(H3) hold and(t,x)∈[0,T)×Rn be fixed. Suppose that u-(·) is an optimal control for Problem (RSOCP), and (X-t,x;u-(·),Y-t,x;u-(·),Z-t,x;u-(·)) is the corresponding optimal state. Let (p(·),q(·),k(·)) be the adjoint processes. Then
(19)〈Hu(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),u-(s),p(s),q(s),aaaaasX-t,x;u-(s)k(s)),u-u-(s)〉≥0,∀u∈U,
a.e. s∈[t,T], P-a.s.Proof.
It is an immediate consequence of [10, Theorem 4.4 of Peng].As we mentioned in the introduction, we can also prove that, under some additional convexity conditions, the above necessary condition in Lemma3 is also sufficient.Lemma 4 (sufficient maximum principle).
Let (H1)–(H3) hold. Suppose thatu-(·) is an admissible control, and (X-t,x;u-(·),Y-t,x;u-(·),Z-t,x;u-(·)) is the corresponding state, with Y(T)=MT⊤X(T), MT∈Rn. Let (p(·),q(·),k(·)) be the adjoint processes. Suppose that the Hamiltonian function H is convex in (x,y,z,u). Then u-(·) is an optimal control for Problem (RSOCP) if it satisfies (19).Proof.
For anyu(·)∈𝒰[t,T] with the corresponding state (Xt,x;u(·),Yt,x;u(·),Zt,x;u(·)). By Remark 1, we have for fixed t∈[0,T],
(20)J(t,x;u-(t))-J(t,x;u(t))=Yt,x;u(t)-Y-t,x;u-(t)=𝔼[Yt,x;u(t)-Y-t,x;u-(t)].
Applying Itô’s formula to 〈Y-t,x;u-(s)-Yt,x;u(s),q(s)〉+〈X-t,x;u-(s)-Xt,x;u(s),p(s)〉, noting (14), (17), and Y-t,x;u-(T)-Yt,x;u(T)=MT⊤(X-t,x;u-(T)-Xt,x;u(T)), p(T)=-MTq(T), we get
(21)𝔼(Y-t,x;u-(T)-Yt,x;u(T))q(T)-𝔼(Y-t,x;u-(t)-Yt,x;u(t))q(t)+𝔼〈X-t,x;u-(T)-Xt,x;u(T),p(T)〉-𝔼〈X-t,x;u-(t)-Xt,x;u(t),p(t)〉=-𝔼[Y-t,x;u-(t)-Yt,x;u(t)]=𝔼∫tT‍{〈Y-t,x;u-(s)-Yt,x;u(s),f-y(s)⊤q(s)〉aaaaaaaaaaaaa+〈Z-t,x;u-(s)-Zt,x;u(s),f-z(s)⊤q(s)〉aaaaaaaaaaaaa-〈f-(s)-f(s,Xt,x;u(s),Yt,x;u(s),aaaaaaaaaaaaaaaaaaaaaaaZt,x;u(s),u(s)Xt,x;u(s),),q(s)f-(s)〉aaaaaaaaaaaaa+〈X-t,x;u-(s)-Xt,x;u(s),-b-x(s)⊤p(s)aaaaaaaaaaaaaa+f-x(s)⊤q(s)-σ-x(s)k(s)X-t,x;u-(s)X-t,x;u-(s)〉aaaaaaaaaaaaa+〈b-(s)-b(s,Xt,x;u(s),u(s)),p(s)〉aaaaaaaaaaaaa+tr[(σ-(s)-σ(s,Xt,x;u(s),u(s)))⊤k(s)]ds=𝔼∫tT‍{H(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),u-(s),aaaaaaaaaaaaap(s),q(s),k(s)X-t,x;u-(s))aaaaaaaaaaaaa-H(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),aaaaaaaaaaaaaaaau(s),p(s),q(s),k(s)Xt,x;u(s)Zt,x;u(s))aaaaaaaaaaaaa-〈Hx(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),aaaaaaaaaaaaaaaaaau-(s),p(s),q(s),k(s)X-t,x;u-Y-t,x;u-),aaaaaaaaaaaaaaaX-t,x;u-(s)-Xt,x;u(s)〉aaaaaaaaaaaaa-〈Hy(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),aaaaaaaaaaaaaaaaaau-(s),p(s),q(s),k(s)Y-t,x;u-X-t,x;u-),aaaaaaaaaaaaaaaY-t,x;u-(s)-Yt,x;u(s)〉aaaaaaaaaaaaa-〈Hz(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),aaaaaaaaaaaaaaaaaaau-(s),p(s),q(s),k(s)Z-t,x;u-Y-t,x;u-),aaaaaaaaaaaaaaaaaZ-t,x;u-(s)-Zt,x;u(s)〉X-t,x;u-(s)}ds.
Since H is convex in (x,y,z,u), then by the maximum condition (19),
(22)J(t,x;u-(t))-J(t,x;u(t))≤𝔼∫tT‍〈Hu(s,X-t,x;u-(s),Y-t,x;u-(s),Z-t,x;u-(s),u-(s),aaaaaaaaaaaaaX-t,x;u-p(s),q(s),k(s)),u-(s)-u(s)〉≤0.
Thus u-(·) is really the optimal control for Problem (RSOCP). The proof is complete.


## 3. Relationship between Maximum Principle and Dynamic Programming

In this section, we investigate the relationship between the maximum principle and dynamic programming. That is, the connection between the value functionV, the generalized Hamiltonian function G, and adjoint processes p, q, k. Our main result is the following.Theorem 5.
Let (H1)–(H3) hold and let(t,x)∈[0,T)×Rn be fixed. Suppose that u-(·) is an optimal control for Problem (RSOCP), and (X-t,x;u-(·),Y-t,x;u-(·),Z-t,x;u-(·)) is the corresponding optimal state. Let (p(·),q(·),k(·)) be the adjoint processes. If V∈C1,2([0,T]×Rn), then
(23)Vs(s,X-t,x;u-(s))=G(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaa-Vx(s,X-t,x;u-(s)),-Vxx(s,X-t,x;u-(s)),u-(s))=maxu∈UG(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaa-Vx(s,X-t,x;u-(s)),-Vxx(s,X-t,x;u-(s)),u),
a.e. s∈[t,T], P-a.s. Furthermore, if V∈C1,3([0,T]×Rn) and Vsx is also continuous, then
(24)p(s)=Vx(s,X-t,x;u-(s))⊤q(s),∀s∈[t,T],P-a.s.,k(s)={Vxx(s,X-t,x;u-(s))⊤σ-(s)+Vx(s,X-t,x;u-(s))⊤aaa×fz(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))(s,X-t,x;u-(s))⊤}q(s),aaaaaaaaaaaaaaaaaaa.e.s∈[t,T],P-a.s.,
where
(25)q(s)=exp{∫ts‍[|fz(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),-Vx(r,X-t,x;u-(r))σ-(r),u-(r))|2fy(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),aaaaaaaaaaaa-Vx(r,X-t,x;u-(r),u-(r))σ-(r),u-(r))aaaaaaaaaa-12|fz(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),aaaaaaaaaaaaaaaaaa-Vx(r,X-t,x;u-(r))aaaaaaaaaaaaaaaaaa×σ-(r),u-(r)(r,X-t,x;u-(r)))(r,X-t,x;u-(r))|2(r,X-t,x;u-(r))]draaaaaaaa+∫ts‍fz(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),aaaaaaaaaaaaaaaaaa-Vx(r,X-t,x;u-(r))aaaaaaaaaaaaaaaaaa×σ-(r),u-(r)(r,X-t,x;u-(r)))dW(r)[|fz(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),-Vx(r,X-t,x;u-(r))σ-(r),u-(r))|2fy(r,X-t,x;u-(r),-V(r,X-t,x;u-(r)),-Vx(r,X-t,x;u-(r),u-(r))σ-(r),u-(r))}.Proof.
Obviously (25) can be obtained via solving the forward SDE in (17) directly. Now let us prove (24). By [15, Theorem 5.4 of Peng], for fixed t∈[0,T), it is easy to obtain
(26)V(s,X-t,x;u-(s))=-Y-t,x;u-(s)=-𝔼[∫sT‍f-(r)dr+ϕ(X-t,x;u-(T))∣ℱst],aaaaaaaaaaaaaaaaaaa∀s∈[t,T],P-a.s.
Define a square-integrable ℱst-martingale
(27)m(s):=-𝔼[∫tT‍f-(r)dr+ϕ(X-Tt,x;u-)∣ℱst],s∈[t,T].
Thus, by the martingale representation theorem (see Yong and Zhou [9]), there exists a unique M(·)∈Lℱ2([t,T];Rd) satisfying
(28)m(s)=m(t)+∫ts‍M(r)dW(r),
where t∈[0,T) is fixed. Then
(29)V(s,X-t,x;u-(s))=-∫sT‍f-(r)dr-∫sT‍M(r)dW(r)+V(T,X-t,x;u-(T)).
On the other hand, applying Itô’s formula to V(s,X-t,x;u-(s)), we obtain
(30)dV(s,X-t,x;u-(s))={(σ-(s)⊤Vxx(s,X-t,x;u-(s))σ-(s))Vs(s,X-t,x;u-(s))+〈Vx(s,X-t,x;u-(s)),b-(s)〉aaaaaaa+12tr(σ-(s)⊤Vxx(s,X-t,x;u-(s))σ-(s))}dsaaaaaaaa+Vx(s,X-t,x;u-(s))⊤σ-(s)dW(s).
Comparing the above two equalities, we conclude that
(31)Vs(s,X-t,x;u-(s))+〈Vx(s,X-t,x;u-(s)),b-(s)〉+12tr(σ-(s)⊤Vxx(s,X-t,x;u-(s))σ-(s))=f-(s),Vx(s,X-t,x;u-(s))⊤σ-(s)=M(s),a.e.s∈[t,T],P-a.s.
However, by the uniqueness of solution to BSDE (3), we have
(32)Y-t,x;u-(s)=-V(s,X-t,x;u-(s)),Z-t,x;u-(s)=-Vx(s,X-t,x;u-(s))⊤σ-(s),a.e.s∈[t,T],P-a.s.
SinceV∈C1,2([0,T]×Rn), it satisfies the generalized HJB equation (7), which implies (23). Also, by (7), we have
(33)0=-Vs(s,X-t,x;u-(s))+G(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),-Vx(s,X-t,x;u-(s)),aaaaaa-Vxx(s,X-t,x;u-(s)),u-(s))≥-Vs(s,x)+G(s,x,-V(s,x),-Vx(s,x),aaaaaaaaaaaaaaaa-Vxx(s,x),u-(s)),∀x∈Rn.
Consequently, if V∈C1,3([0,T]×Rn) and Vsx is also continuous, then
(34)∂∂x{-Vs(s,x)+G(s,x,-V(s,x),-Vx(s,x),aaaaaaaaaaaaaaa-Vxx(s,x),u-(s))}|x=X-t,x;u-(s)=0,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa∀s∈[t,T].
This is equivalent to (recall (8)), for all s∈[t,T],
(35)0=-Vsx(s,X-t,x;u-(s))-Vxx(s,X-t,x;u-(s))b-(s)-b-x(s)⊤Vx(s,X-t,x;u-(s))-12tr(σ-(s)⊤Vxxx(s,X-t,x;u-(s))σ-(s))-σ-x(s)⊤Vxx(s,X-t,x;u-(s))σ-(s)+fx(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s)(s,X-t,x;u-(s)))-fy(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))×Vx(s,X-t,x;u-(s))-fz(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))×[Vxx(s,X-t,x;u-(s))σ-(s)+Vx(s,X-t,x;u-(s))σ-x(s)],
where
(36)tr(σ-⊤Vxxxσ-):=(tr(σ-⊤((Vx)1)xxσ-),…,tr(σ-⊤((Vx)n)xxσ-))⊤,
with ((Vx)1,…,(Vx)n)⊤=Vx.
On the other hand, applying Itô’s formula toVx(s,X-t,x;u-(s)), we get
(37)dVx(s,X-t,x;u-(s))=-{[Vxx(s,X-t,x;u-(s))σ-(s)+Vx(s,X-t,x;u-(s))σ-x(s)]b-x(s)⊤Vx(s,X-t,x;u-(s))+σ-x(s)⊤aaaaaaaa×Vxx(s,X-t,x;u-(s))σ-(s)aaaaaaaaa-fx(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))aaaaaaaaa+fy(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))aaaaaaaa×Vx(s,X-t,x;u-(s))aaaaaaaaa+fz(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))aaaaaaaaa×[Vxx(s,X-t,x;u-(s))σ-(s)aaaaaaaaa+Vx(s,X-t,x;u-(s))σ-x(s)]Vx(s,X-t,x;u-(s))}ds+Vxx(s,X-t,x;u-(s))⊤σ-(s)dW(s).
Note that Vx(T,X-t,x;u-(T))=-ϕx(X-t,x;u-(T)). Applying again Itô’s formula to Vx(s,X-t,x;u-(s))⊤q(s), we have
(38)dVx(s,X-t,x;u-(s))⊤q(s)={(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),-Vx(s,X-t,x;u-(s))σ-(s),u-(s))⊤q(s)-b-x(s)⊤Vx(s,X-t,x;u-(s))q(s)aaaaaaa+fx(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))⊤q(s)aaaaaaa-σ-x(s)⊤[Vxx(s,X-t,x;u-(s))⊤σ-(s)aaaaaaaaaaaaaaaaa+Vx(s,X-t,x;u-(s))⊤aaaaaaaaaaaaaaaaa×fz(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaaaaaaaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-s)Vxx(s,X-t,x;u-(s))⊤]q(s)(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),-Vx(s,X-t,x;u-(s))σ-(s),u-(s))⊤q(s)}ds+{Vxx(s,X-t,x;u-(s))⊤σ-(s)+Vx(s,X-t,x;u-(s))⊤aaaaaaa×fz(s,X-t,x;u-(s),-V(s,X-t,x;u-(s)),aaaaaaa-Vx(s,X-t,x;u-(s))σ-(s),u-(s))(s,X-t,x;u-(s))⊤}×q(s)dW(s).
Hence, by the uniqueness of the solutions to the adjoint equation (17), we obtain (24). The proof is complete.


## 4. Applications to Financial Portfolio Optimization

In this section, we consider an LQ recursive utility portfolio optimization problem in the financial engineering. In this problem, the optimal portfolio in the state feedback form is obtained by both maximum principle and dynamic programming approaches, and the relations which we obtained in Theorem5 are illustrated explicitly.Suppose the investors have two kinds of securities in the market for possible investment choice.(i)
A risk-free security (e.g., a bond), where the priceS0(t) at time t is given by
(39)dS0(t)=ρtS0(t)dt,S0(0)>0,
here ρt is a bounded deterministic function.(ii)
A risky security (e.g., a stock), where the priceS1(t) at time t is given by
(40)dS1(t)=μtS1(t)dt+σtS1(t)dW(t),S1(0)>0,
here W(·) is a 1-dimensional Brownian motion and μt,σt≠0 are bounded deterministic functions with μt>ρt.Letu(t) denote the total market value of the investor’s wealth invested in the risky security which we call portfolio. Given the initial wealth Xu(0)=X0≥0, combining (39) and (40), we can get the following wealth dynamics:
(41)dXu(t)=[ρtXu(t)+(μt-ρt)u(t)]dt+σtu(t)dW(t),t≥0,Xu(0)=X0.
We denote by Uad the set of admissible portfolios valued in U=R.For any given initial wealthX0>0, Kohlmann and Zhou [28] discussed a mean-variance portfolio optimization problem. That is, the investor’s object is to find an admissible portfolio u*(t) which minimizes the variance Var[Xu(T)]:=𝔼[(Xu(T)-𝔼[Xu(T)])2] at some future time T>0 under the condition that 𝔼[Xu(T)]=A for some given A∈R. Using the Lagrange multiplier method, we know that it is equivalent to study the following problem:
(42)supu(·)∈Uad𝔼[-12(Xu(T)-a)2],
where some a∈R is given. Using the completion of squares technique, the study of [28] obtained an optimal portfolio in the state feedback form by some stochastic Riccati equation and BSDE. The optimal value function was also obtained.In this paper, we generalize the above mean-variance portfolio optimization problem to a recursive utility portfolio optimization problem. The recursive utility means that the utility at timet is a function of the future utility (in this section, we do not consider the consumption). In fact, in our framework, the recursive utility can be assumed to satisfy some controlled BSDE.We consider a small investor, endowed with initial wealthX0>0, who chooses at each time t his/her portfolio u(t). The investor wants to choose an optimal portfolio u*(·)∈Uad to maximize the following recursive utility functional with generator:
(43)f(t,u,x,y)=ρtx+(μt-ρt)u-βy,
where constant β≥0.Remark 6.
In fact, the recursive utility functional (43) defined above stands for some standard additive utility of recursive type. It is a meaningful and nontrivial generalization of the classical standard additive utility and has many applications in mathematical economics and mathematical finance. For more details about utility functions, see Duffie and Epstein [2], Section 1.4 of El Karoui et al. [3] or Schroder and Skiadas [4].
More precisely, for anyu(·)∈Uad, the investor’s utility functional is defined by
(44)J-(u(·)):=Yu(t)|t=0,
where
(45)Yu(t)≔𝔼[∫tT‍-12(Xu(T)-a)2aaaaa+∫tT‍[ρsXu(s)+(μs-ρs)u(s)-βYu(s)]ds∣ℱt],aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa∀t∈[0,T].
In fact, in our framework, the wealth processXu(·) and recursive utility process Yu(·) can be regarded as the solution to the following controlled FBSDE:
(46)dXu(t)=[ρtXu(t)+(μt-ρt)u(t)]dt+σtu(t)dW(t),-dYu(t)=[ρtXu(t)+(μt-ρt)u(t)-βYu(t)]dt-Zu(t)dW(t),t∈[0,T],Xu(0)=X0,Yu(T)=-12(Xu(T)-a)2,
and our portfolio optimization problem can be rewritten as (denoting J=-J-)
(47)J(u*(·))=infu(·)∈UadJ(u(·)).
Since we are going to involve the dynamic programming in treating the above problem, we will adopt the formulation as in Section2. Let T>0 be given. For any (t,x)∈[0,T)×R, consider the following controlled FBSDE:
(48)dXt,x;u(s)=[ρsXt,x;u(s)+(μs-ρs)u(s)]ds+σsu(s)dW(s),-dYt,x;u(s)=[ρsXt,x;u(s)+(μs-ρs)u(s)-βYt,x;u(s)]ds-Zt,x;u(s)dW(s),s∈[t,T],Xt,x;u(t)=x,Yt,x;u(T)=-12(Xt,x;u(T)-a)2.
And our recursive utility portfolio optimization problem is to find an optimal portfolio u*(·)∈Uad to minimize the recursive utility functional J(t,x;u(·)):=-Yt,x;u(s)|s=t. We define the value function as
(49)V(t,x):=J(t,x;u*(·))=infu(·)∈UadJ(t,x;u(·)).
We can check that all the assumptions in Section2 are satisfied. Then we can use both the dynamic programming (Lemma 2) and maximum principle (Lemmas 3 and 4) approaches to solve the above problem (49).


### 4.1. Maximum Principle Approach

In this case, the Hamiltonian function (18) reduces to
(50)H(s,x,y,z,u,p,q,k)=p[ρsx+(μs-ρs)u]-q[ρsx+(μs-ρs)u-βy]+kσsu,
and the adjoint equation (17) writes
(51)-dp(s)=ρs[p(s)-q(s)]ds-k(s)dW(s),dq(s)=-βq(s)ds,s∈[t,T],p(T)=[Xt,x;u(T)-a]q(T),q(t)=1.
Noting that in this case the adjoint process q(·) reduced to a deterministic function because our generator f does not contain the process z(·). We immediately have
(52)q(s)=e-β(s-t),s∈[t,T].
Due to the terminal condition of (51), we try a process p(·) of the form
(53)p(s)=[ϕsXt,x;u(s)+ψs]e-β(s-t),s∈[t,T],
where ϕs, ψs are deterministic differentiable functions. Applying Itô’s formula to (53), we have
(54)dp(s)=e-β(s-t)[(ϕ˙s+(ρs-β)ϕs)Xt,x;u(s)aaaaaaaaa+ϕs(μs-ρs)u(s)+ψ˙s-βψs(ϕ˙s+(ρs-β)ϕs)]ds+e-β(s-t)ϕsσsu(s)dW(s).
Comparing (51) with (54), noting (52) and (53), we get
(55)(ϕ˙s+(ρs-β)ϕs)Xt,x;u(s)+ϕs(μs-ρs)u(s)+ψ˙s-βψs=-ρs(ϕsXt,x;u(s)+ψs-1),(56)k(s)=ϕsσsu(s)e-β(s-t).Letu*(·) be a candidate optimal portfolio, (Xt,x;u*(·),Yt,x;u*(·),Zt,x;u*(·)) the corresponding solution to controlled FBSDE (48) with corresponding solution (p*(·),q*(·),k*(·)) to the adjoint equation (51). Now the Hamiltonian function (50) is
(57)H(s,Xt,x;u*(s),Yt,x;u*(s),Zt,x;u*(s),u,p*(s),q*(s),k*(s))=p*(s)[ρsXt,x;u*(s)+(μs-ρs)u]-q*(s)[ρsXt,x;u*(s)+(μs-ρs)u-βYt,x;u*(s)]+k*(s)σsu.
Since this is a linear expression of u, by the maximum condition (19), we have
(58)(μs-ρs)(p*(s)-q*(s))+σsk*(s)=0.
Substituting (52), (53) and (56) into (58), we can get
(59)u*(s)=(ρs-μs)(ϕsXt,x;u*(s)+ψs-1)ϕsσs2.On the other hand, (55) gives
(60)u*(s)=(ϕ˙s+2ρsϕs-βϕs)Xt,x;u*(s)+ψ˙s+(ρs-β)ψs-ρsϕs(ρs-μs).Combining (59) and (60) (noting the terminal condition in (51)), we get
(61)ϕ˙s=[(ρs-μs)2σs2-2ρs+β]ϕs,ϕT=1,(62)ψ˙s=[(ρs-μs)2σs2-ρs+β]ψs-(ρs-μs)2σs2+ρs,ψT=-a.
The solutions to these equations are
(63)ϕs=e-∫sT‍[(ρr-μr)2/σs2-2ρr+β]dr,(64)ψs=e-∫sT‍[(ρr-μr)2/σr2-ρr+β]dr×{∫sT‍[(ρr-μr)2σr2-ρr]e∫sT‍[(ρλ-μλ)2/σλ2-ρλ+β]dλdr-a}.
With this choice of ϕs and ψs, the processes
(65)q*(s)=e-β(s-t),p*(s)=[ϕsXt,x;u*(s)+ψs]e-β(s-t),k*(s)=ϕsσsu*(s)e-β(s-t)
satisfying the adjoint equation (51) with u*(s) given by (59). With this choice of u*(s), the maximum condition (19) of Lemma 3 holds. Moreover, we can check that all conditions in Lemma 4 hold, then u*(s) given by (59) is really the optimal control.Finally, lett=0, then we can solve the initial problem (47) and give the explicit optimal portfolio in the state feedback form.Theorem 7.
The optimal solutionu*(·) of our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by
(66)u*(s,X*)=(ρs-μs)(ϕsX*+ψs-1)ϕsσs2,
for s∈[0,T], where ϕs, ψs are given by (63) and (64), respectively.


### 4.2. Dynamic Programming Approach

In this case, the value functionV should satisfy the following generalized HJB equation:
(67)-Vt(t,x)+supu∈UG(t,x,-V(t,x),-Vx(t,x),aaaaaaaaaaaaaaaa-Vxx(t,x),u)=0,aaaaaaaaaaaaaaaaa(t,x)∈[0,T]×Rn,V(T,x)=12(x-a)2,x∈Rn,
where the generalized Hamiltonian function (8) is
(68)G(t,x,-V(t,x),-Vx(t,x),-Vxx(t,x),u)=-12Vxx(t,x)σs2u2-Vx(t,x)[ρtx+(μt-ρt)u]+ρtx+(μt-ρt)u+βV(t,x).We conjecture thatV(t,x) is quadratic in x, namely,
(69)V(t,x)=12Ptx2+φtx+λt,
for some deterministic differentiable functions Pt, φt, and λt with
(70)PT=1,φT=-a,λT=12a2.
Substituting (69) into (68) and using completion of squares repeatedly, we get
(71)G(t,x,-V(t,x),-Vx(t,x),-Vxx(t,x),u)=-12Ptσt2[u(t)-(ρt-μt)(Ptx+φt-1)Ptσt2]2+[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt≤[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt,
provided that Pt>0 for all t∈[0,T], which we will prove later. Then we see that the optimal state feedback portfolio is given by
(72)u*(t,X*)=(ρt-μt)(PtX*+φt-1)Ptσt2.
In addition, the generalized HJB equation (67) now reads
(73)12P˙tx2+φ˙tx+λ˙t=[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt.
Then noting (70), by comparing the quadratic terms and linear terms in x, we recover (61) and (62), respectively. That is to say, we have that Pt coincides with ϕt and φt coincides with ψt. Then by (63), we have ϕt>0, for all t∈[0,T] as expected before. And also we have
(74)λ˙t=-(ρt-μt)2ψtϕtσt2+βλt,λT=12a2.
The solution to (74) is
(75)λt=e-β(T-t){∫tT‍[(ρr-μr)2ψrϕrσr2-ρr]eβ(T-r)dr+12a2}.
Then the value function
(76)V(t,x)=12ϕtx2+ψtx+λt,
where ϕt, ψt, and λt are determined by (61), (62), and (75), respectively. By Lemma 2, we have proved the followingTheorem 8.
The optimal solutionu*(·) for our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by
(77)u*(t,X*)=(ρt-μt)(ϕtX*+ψt-1)ϕtσt2,
for t∈[0,T], and the value function is given by (76), where ϕt, ψt, and λt are determined by (61), (62), and (75), respectively.


### 4.3. Relationship

We now can explicitly illustrate the relationships in Theorem5. In fact, relationship (23) is obvious from (67). And (65) is exactly the relationship given in (25) and (24).


## 4.1. Maximum Principle Approach

In this case, the Hamiltonian function (18) reduces to
(50)H(s,x,y,z,u,p,q,k)=p[ρsx+(μs-ρs)u]-q[ρsx+(μs-ρs)u-βy]+kσsu,
and the adjoint equation (17) writes
(51)-dp(s)=ρs[p(s)-q(s)]ds-k(s)dW(s),dq(s)=-βq(s)ds,s∈[t,T],p(T)=[Xt,x;u(T)-a]q(T),q(t)=1.
Noting that in this case the adjoint process q(·) reduced to a deterministic function because our generator f does not contain the process z(·). We immediately have
(52)q(s)=e-β(s-t),s∈[t,T].
Due to the terminal condition of (51), we try a process p(·) of the form
(53)p(s)=[ϕsXt,x;u(s)+ψs]e-β(s-t),s∈[t,T],
where ϕs, ψs are deterministic differentiable functions. Applying Itô’s formula to (53), we have
(54)dp(s)=e-β(s-t)[(ϕ˙s+(ρs-β)ϕs)Xt,x;u(s)aaaaaaaaa+ϕs(μs-ρs)u(s)+ψ˙s-βψs(ϕ˙s+(ρs-β)ϕs)]ds+e-β(s-t)ϕsσsu(s)dW(s).
Comparing (51) with (54), noting (52) and (53), we get
(55)(ϕ˙s+(ρs-β)ϕs)Xt,x;u(s)+ϕs(μs-ρs)u(s)+ψ˙s-βψs=-ρs(ϕsXt,x;u(s)+ψs-1),(56)k(s)=ϕsσsu(s)e-β(s-t).Letu*(·) be a candidate optimal portfolio, (Xt,x;u*(·),Yt,x;u*(·),Zt,x;u*(·)) the corresponding solution to controlled FBSDE (48) with corresponding solution (p*(·),q*(·),k*(·)) to the adjoint equation (51). Now the Hamiltonian function (50) is
(57)H(s,Xt,x;u*(s),Yt,x;u*(s),Zt,x;u*(s),u,p*(s),q*(s),k*(s))=p*(s)[ρsXt,x;u*(s)+(μs-ρs)u]-q*(s)[ρsXt,x;u*(s)+(μs-ρs)u-βYt,x;u*(s)]+k*(s)σsu.
Since this is a linear expression of u, by the maximum condition (19), we have
(58)(μs-ρs)(p*(s)-q*(s))+σsk*(s)=0.
Substituting (52), (53) and (56) into (58), we can get
(59)u*(s)=(ρs-μs)(ϕsXt,x;u*(s)+ψs-1)ϕsσs2.On the other hand, (55) gives
(60)u*(s)=(ϕ˙s+2ρsϕs-βϕs)Xt,x;u*(s)+ψ˙s+(ρs-β)ψs-ρsϕs(ρs-μs).Combining (59) and (60) (noting the terminal condition in (51)), we get
(61)ϕ˙s=[(ρs-μs)2σs2-2ρs+β]ϕs,ϕT=1,(62)ψ˙s=[(ρs-μs)2σs2-ρs+β]ψs-(ρs-μs)2σs2+ρs,ψT=-a.
The solutions to these equations are
(63)ϕs=e-∫sT‍[(ρr-μr)2/σs2-2ρr+β]dr,(64)ψs=e-∫sT‍[(ρr-μr)2/σr2-ρr+β]dr×{∫sT‍[(ρr-μr)2σr2-ρr]e∫sT‍[(ρλ-μλ)2/σλ2-ρλ+β]dλdr-a}.
With this choice of ϕs and ψs, the processes
(65)q*(s)=e-β(s-t),p*(s)=[ϕsXt,x;u*(s)+ψs]e-β(s-t),k*(s)=ϕsσsu*(s)e-β(s-t)
satisfying the adjoint equation (51) with u*(s) given by (59). With this choice of u*(s), the maximum condition (19) of Lemma 3 holds. Moreover, we can check that all conditions in Lemma 4 hold, then u*(s) given by (59) is really the optimal control.Finally, lett=0, then we can solve the initial problem (47) and give the explicit optimal portfolio in the state feedback form.Theorem 7.
The optimal solutionu*(·) of our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by
(66)u*(s,X*)=(ρs-μs)(ϕsX*+ψs-1)ϕsσs2,
for s∈[0,T], where ϕs, ψs are given by (63) and (64), respectively.


## 4.2. Dynamic Programming Approach

In this case, the value functionV should satisfy the following generalized HJB equation:
(67)-Vt(t,x)+supu∈UG(t,x,-V(t,x),-Vx(t,x),aaaaaaaaaaaaaaaa-Vxx(t,x),u)=0,aaaaaaaaaaaaaaaaa(t,x)∈[0,T]×Rn,V(T,x)=12(x-a)2,x∈Rn,
where the generalized Hamiltonian function (8) is
(68)G(t,x,-V(t,x),-Vx(t,x),-Vxx(t,x),u)=-12Vxx(t,x)σs2u2-Vx(t,x)[ρtx+(μt-ρt)u]+ρtx+(μt-ρt)u+βV(t,x).We conjecture thatV(t,x) is quadratic in x, namely,
(69)V(t,x)=12Ptx2+φtx+λt,
for some deterministic differentiable functions Pt, φt, and λt with
(70)PT=1,φT=-a,λT=12a2.
Substituting (69) into (68) and using completion of squares repeatedly, we get
(71)G(t,x,-V(t,x),-Vx(t,x),-Vxx(t,x),u)=-12Ptσt2[u(t)-(ρt-μt)(Ptx+φt-1)Ptσt2]2+[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt≤[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt,
provided that Pt>0 for all t∈[0,T], which we will prove later. Then we see that the optimal state feedback portfolio is given by
(72)u*(t,X*)=(ρt-μt)(PtX*+φt-1)Ptσt2.
In addition, the generalized HJB equation (67) now reads
(73)12P˙tx2+φ˙tx+λ˙t=[(μt-ρt)2Pt2σt2+12βPt-ρtPt]x2+[(ρt-μt)2σt2(φt-1)+βφt-ρtφt+ρt]x-(ρt-μt)2φtPtσt2+βλt.
Then noting (70), by comparing the quadratic terms and linear terms in x, we recover (61) and (62), respectively. That is to say, we have that Pt coincides with ϕt and φt coincides with ψt. Then by (63), we have ϕt>0, for all t∈[0,T] as expected before. And also we have
(74)λ˙t=-(ρt-μt)2ψtϕtσt2+βλt,λT=12a2.
The solution to (74) is
(75)λt=e-β(T-t){∫tT‍[(ρr-μr)2ψrϕrσr2-ρr]eβ(T-r)dr+12a2}.
Then the value function
(76)V(t,x)=12ϕtx2+ψtx+λt,
where ϕt, ψt, and λt are determined by (61), (62), and (75), respectively. By Lemma 2, we have proved the followingTheorem 8.
The optimal solutionu*(·) for our recursive utility portfolio optimization problem (47), when the wealth dynamics obeys (41), is given in the state feedback form by
(77)u*(t,X*)=(ρt-μt)(ϕtX*+ψt-1)ϕtσt2,
for t∈[0,T], and the value function is given by (76), where ϕt, ψt, and λt are determined by (61), (62), and (75), respectively.


## 4.3. Relationship

We now can explicitly illustrate the relationships in Theorem5. In fact, relationship (23) is obvious from (67). And (65) is exactly the relationship given in (25) and (24).


## 5. Concluding Remarks

In this paper, we have studied the relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems. Under certain differentiability conditions, we give relations among the adjoint processes, the generalized Hamiltonian function, and the value function. A linear quadratic recursive utility portfolio optimization problem in the financial market is discussed as an explicitly illustrated example of our result.An interesting and challenging problem remains open. For the stochastic recursive optimal control problem, what is the relationship between maximum principle and dynamic programming without the illusory differentiability conditions on the value function? This problem may be solved in the framework of nonsmooth analysis. Viscosity solution theory is certainly a nice tool (e.g., see Yong and Zhou [9]). A new result on stochastic verification theorem for forward-backward controlled system using viscosity solutions has been published very recently by Zhang [29]. However, at this moment, we do not have publishable results for the relationship within the framework of viscosity solutions. We hope to address this problem in the future work.



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*Source: 285241-2013-02-21.xml*