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

Na Li,Zhiyong Yu

doi:10.1186/s13662-015-0439-1

Na Li, Zhiyong Yu

Open Access

https://doi.org/10.1186/s13662-015-0439-1

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Abstract

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.

Highlights

1 Introduction This paper is concerned with a kind of linear-quadratic stochastic optimal control (LQ SOC) problems and linear-quadratic nonzero-sum stochastic differential game (LQ NZSSDG) problems for forward-backward systems with random jumps
For the LQ NZSSDG problems, the controlled system is given by the following controlled linear forward-backward stochastic differential equation with Poisson process (FBSDE):
Inspired by the idea of stochastic maximum principle in optimal control theory and Hamadène’s transform, both the LQ SOC problem and the LQ NZSSDG problem are closed linked to a kind of coupled FBSDEs involving Poisson jumps, which is out of scope of the existing results in the FBSDEs theory

Summary

Introduction

This paper is concerned with a kind of linear-quadratic stochastic optimal control (LQ SOC) problems and linear-quadratic nonzero-sum stochastic differential game (LQ NZSSDG) problems for forward-backward systems with random jumps. For the LQ NZSSDG problems, the controlled system is given by the following controlled linear forward-backward stochastic differential equation with Poisson process (FBSDE): For the forward controlled system, Hamadène [ ] linked an LQ nonzero-sum stochastic differential game problem to a linear coupled FBSDE by virtue of changing variables, and he gave an existence result for Nash equilibrium points of the game problem.

Results

Conclusion