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Constrained zero-sum LQ differential games for jump-diffusion systems with regime switching and random coefficients

Yanyan Tang, Xu Li, Jie Xiong

Abstract

This paper investigates a cone-constrained two-player zero-sum stochastic linear-quadratic (SLQ) differential game for stochastic differential equations (SDEs) with regime switching and random coefficients driven by a jump-diffusion process. Under the uniform convexity-concavity (UCC) condition, we establish the open-loop solvability of the game and characterize the open-loop saddle point via the forward-backward stochastic differential equations (FBSDEs). However, since both controls are constrained, the classical four-step scheme fails to provide an explicit expression for the saddle point. To overcome this, by employing Meyer's Itô formula together with the method of completing the square, we derive a closed-loop representation for the open-loop saddle point based on solutions to a new kind of multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs). Furthermore, for a special case, we prove the existence of solutions to IESREJs.

Constrained zero-sum LQ differential games for jump-diffusion systems with regime switching and random coefficients

Abstract

This paper investigates a cone-constrained two-player zero-sum stochastic linear-quadratic (SLQ) differential game for stochastic differential equations (SDEs) with regime switching and random coefficients driven by a jump-diffusion process. Under the uniform convexity-concavity (UCC) condition, we establish the open-loop solvability of the game and characterize the open-loop saddle point via the forward-backward stochastic differential equations (FBSDEs). However, since both controls are constrained, the classical four-step scheme fails to provide an explicit expression for the saddle point. To overcome this, by employing Meyer's Itô formula together with the method of completing the square, we derive a closed-loop representation for the open-loop saddle point based on solutions to a new kind of multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs). Furthermore, for a special case, we prove the existence of solutions to IESREJs.
Paper Structure (5 sections, 7 theorems, 107 equations)

This paper contains 5 sections, 7 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Open-loop solvability
  4. Stochastic Riccati equations and open-loop saddle-point
  5. Solvability of the ECSREJs

Key Result

Theorem 3.1

Let Assumptions A1 hold and let the initial pair $(\xi, i)\in L(\Omega, {\fam\msbmfam R})\times{\cal M}$ be given. If for any $u\in{\cal U}$, $J(0,i; u)>0, i\in{\cal M}$, then a process $u^*\in{\cal U}$ is an open-loop optimal control of Problem (CSLQJ) if and only if the following condition holds: where $(X^*, Y^*,Z^*, K^*, \Gamma^* )\in S_{{\fam\msbmfam F}}^2(0,T; {\fam\msbmfam R})\times S_{{\f

Theorems & Definitions (20)

  • Definition 3.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.1
  • Remark 4.1
  • ...and 10 more