Recursive Optimal Control Problem Research Articles

Maximum principle for forward–backward stochastic control system under [formula omitted]-expectation and relation to dynamic programming

In this paper, based on the theory of stochastic differential equations on a sublinear expectation space (Ω,H,Eˆ), we develop a stochastic maximum principle for a general forward–backward stochastic control system under G-expectation. Under some convexity assumptions, we also obtain sufficient conditions for the optimality. Furthermore, relations between the adjoint processes and the value function for stochastic recursive optimal control problems are given. Finally, applications of our main results to the recursive utility portfolio optimization problem in financial market are discussed.

Recursive stochastic linear-quadratic optimal control and nonzero-sum differential game problems with random jumps

This paper presents an existence and uniqueness result for a kind of forward-backward stochastic differential equations (FBSDEs for short) driven by Brownian motion and Poisson process under some monotonicity conditions. By virtue of the conclusion of FBSDEs, we solve a linear-quadratic stochastic optimal control problem for forward-backward stochastic systems with random jumps. Moreover, we also solve a linear-quadratic nonzero-sum stochastic differential game problem. We obtain explicit forms of the unique optimal control and the unique Nash equilibrium point, respectively.

Stochastic recursive optimal control problem with time delay and applications

This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.

Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lévy processes

In this paper, we study the stochastic maximum principle for optimal control problem of anticipated forward-backward system with delay and Levy processes as the random disturbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and Levy processes (AFBSDEDLs), we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs’ preliminary result with certain classical convex variational techniques, the corresponding maximum principle is proved.

A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

Sobolev Weak Solutions of the Hamilton--Jacobi--Bellman Equations

This paper is concerned with the Sobolev weak solutions of the Hamilton--Jacobi--Bellman (HJB) equations. These equations are derived from the dynamic programming principle in the study of stochastic optimal control problems. Adopting the Doob--Meyer decomposition theorem as one of the main tools, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. In the recursive optimal control problem, the cost function is described by the solution of a backward stochastic differential equation (BSDE). This problem has a practical background in economics and finance. We prove that the value function is the unique Sobolev weak solution of the related HJB equation by virtue of the nonlinear Doob--Meyer decomposition theorem introduced in the study of BSDEs.

A Classical Stochastic Verification Theorem for Stochastic Recursive Optimization Problems

In this paper, we study a stochastic recursive optimal control problem in which the system is governed by a forward-backward stochastic differential equation. Under mild assumptions, a classical stochastic verification theorem is derived.

Relationship between Maximum Principle and Dynamic Programming for Stochastic Recursive Optimal Control Problems and Applications

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.

Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions

SUMMARYThis paper is concerned with the relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions. Under the assumption that the value function is smooth, 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 market is discussed to show the applications of the main result. Copyright © 2012 John Wiley & Sons, Ltd.

Maximum principle of recursive optimal control problem for forward-backward stochastic delayed system with Poisson jumps

This paper is concerned with the recursive optimal control problem for forward-backward stochastic delayed system with Poisson jumps. By virtue of classical spike variational approach, duality method and anticipated backward stochastic differential equation with Poisson jumps, we prove the maximum principle of the optimal control for this problem. Both necessary and sufficient conditions of optimality are obtained.

Dynamic programming principle for stochastic recursive optimal control problem with delayed systems

In this paper, we study one kind of stochastic recursive optimal control problem for the systems described by stochastic differential equations with delay (SDDE). In our framework, not only the dynamics of the systems but also the recursive utility depend on the past path segment of the state process in a general form. We give the dynamic programming principle for this kind of optimal control problems and show that the value function is the viscosity solution of the corresponding infinite dimensional Hamilton-Jacobi-Bellman partial differential equation.

Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

The Maximum Principles for Stochastic Recursive Optimal Control Problems Under Partial Information

A maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables. Such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal-agent problems. By virtue of a classical spike variational approach and a filtering technique, the maximum principle is obtained, and the related adjoint processes are characterized by the solutions of forward-backward stochastic differential equations in finite-dimensional spaces. Then our theoretical results are applied to study a partially observed linear-quadratic recursive optimal control problem. In addition, for the case with initial and terminal state constraints, the corresponding maximum principle is also obtained by using Ekeland's variational principle.

Kalman–Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

This paper is concerned with Kalman–Bucy filtering problems of a forward and backward stochastic system which is a Hamiltonian system arising from a stochastic optimal control problem. There are two main contributions worthy pointing out. One is that we obtain the Kalman–Bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation. The other is that we develop a backward separation technique, which is different to Wonham's separation theorem, to study a partially observed recursive optimal control problem. This new technique can also cover some more general situation such as a partially observed linear quadratic non-zero sum differential game problem is solved by it. We also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases.

Dynamic Programming Principle for One Kind of Stochastic Recursive Optimal Control Problem and Hamilton–Jacobi–Bellman Equation

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton–Jacobi–Bellman equation.

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions