Non-zero sum differential games of anticipated forward-backward stochastic differential delayed equations under partial information and application

Yi Zhuang

doi:10.1186/s13662-017-1438-1

Abstract

This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach. We study a linear-quadratic system under partial information and present an explicit form of the Nash equilibrium point. We derive the filtering equations and prove the existence and uniqueness for the Nash equilibrium point. As an application, we solve a time-delayed pension fund management problem with nonlinear expectation to measure the risk and obtain the explicit solution.

Highlights

The general nonlinear backward stochastic differential equations (BSDEs) were first developed by Pardoux and Peng [1] and have been widely applied in the optimal control, mathematical finance, and related fields
The classical Black-Scholes option pricing formula in the financial market can be deduced by virtue of the BSDE theory
If a BSDE is coupled with a forward stochastic differential equation (SDE), it is called the forward-backward stochastic differential equation (FBSDE)

Summary

Introduction

The general nonlinear backward stochastic differential equations (BSDEs) were first developed by Pardoux and Peng [1] and have been widely applied in the optimal control, mathematical finance, and related fields (see Peng [2, 3], Karoui et al [4]). Instead of complete information, there are many cases where the controller can only obtain partial information, reflecting in mathematics that the control variable is adapted to a smaller filtration Based on this phenomenon, Xiong and Zhou [27] dealt with a mean-variance problem in the financial market that the investor’s optimal portfolio is only based on the stock process he observed. Theorem 2.1 If v1(·) and v2(·) are admissible controls and assumption H1 holds, AFBSDDE (1) admits a unique solution (x(·), y(·), z(·), z(·)) ∈ L2F (–δ, T; Rn) × L2F (0, T + δ; Rm) × L2F (0, T ; Rm×d) × L2F (0, T ; Rm×d ) The players have their own preferences which are described by the following cost functionals: T. We aim to establish the necessary and sufficient condition for the Nash equilibrium point subject to this game problem

Maximum principle

R–2 1 Bτ2 B2 R–2 1 Bτ2

R–1 1 B1 R–1 1

Conclusions