Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality

TL;DR

This paper studies a Dynkin game involving stochastic differential equations with random coefficients, establishing a connection to backward stochastic partial differential variational inequalities and characterizing Nash equilibria through strong solutions.

Contribution

It generalizes Krylov estimates and Itô-Kunita-Wentzell's formula for non-Markov processes, and proves existence, uniqueness, and properties of solutions to the associated BSPDVI.

Findings

Established a generalized Krylov estimate for non-Markov processes.

Proved existence and uniqueness of solutions to the BSPDVI.

Characterized Nash equilibrium via strong solutions of the BSPDVI.

Abstract

A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tang's maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a Markov process to that of a non-Markov process, and establish a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be a random field of It\^o's type which takes values in a suitable Sobolev space. We then prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a backward stochastic partial differential variational inequality (BSPDVI, for short) with two obstacles. We obtain the existence and uniqueness result and a comparison theorem for strong solution of the BSPDVI. Moreover, we study the monotonicity on the strong solution of the BSPDVI by the comparison theorem for BSPDVI and define the free boundaries. Finally, we identify the counterparts for an optimal stopping time problem as a special Dynkin game.