Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations^*

¹ School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China.
² School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China.
³ School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China.

^** Corresponding author: weiqm100@nenu.edu.cn

Received: 28 July 2019
Accepted: 19 October 2020

Abstract

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times t and states x to stopping times τ and random variables η ∈ L²(Ω, 𝓕τ, P; ℝ). The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.