A stackelberg game of backward stochastic differential equations with partial information

Zheng Yueyang; Shi Jingtao

doi:10.3934/mcrf.2020047

Article Contents

2021, Volume 11, Issue 4: 797-828. Doi: 10.3934/mcrf.2020047 open access

This issue Previous Article Strict dissipativity for discrete time discounted optimal control problems Next Article Stochastic maximum principle for problems with delay with dependence on the past through general measures

A stackelberg game of backward stochastic differential equations with partial information

Zheng Yueyang^, and
Shi Jingtao

School of Mathematics, Shandong University, Jinan 250100, China

^* Corresponding author: Jingtao Shi
^* Corresponding author: Jingtao Shi

Received: November 30, 2019

Revised: May 31, 2020

Early access: November 2020

Published: December 2021

This work is financially supported by National Key R & D Program of China (2018YFB1305400) and National Natural Science Foundations of China (11971266, 11831010, 11571205)

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Abstract

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$ \sigma $-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE. Four high-dimensional Riccati equations are introduced to represent the state estimate feedback for the optimal control of the leader. Theoretic results are applied to a pension fund management problem of two players in the financial market.

Keywords:

Stackelberg game,
backward stochastic differential equation,
partial information,
maximum principle,
linear-quadratic control.

Mathematics Subject Classification: Primary: 93E20, 49K45, 49N10; Secondary: 49N70, 60H10.

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