Mixed generalized Dynkin game and stochastic control in a Markovian framework

TL;DR

This paper introduces a new mixed generalized Dynkin game and stochastic control framework within a Markovian setting, analyzing both Borelian and continuous terminal reward functions using advanced stochastic analysis tools.

Contribution

It establishes weak and strong dynamic programming principles for the problem, including new results on DRBSDEs and continuity of the value function, linking to Hamilton-Jacobi-Bellman inequalities.

Findings

Established weak dynamic programming principle using refined stochastic analysis.

Proved the value function's continuity in time for the continuous reward case.

Linked the mixed problem to generalized Hamilton-Jacobi-Bellman variational inequalities.

Abstract

We introduce a mixed {\em generalized} Dynkin game/stochastic control with ${\cal E}^f$-expectation in a Markovian framework. We study both the case when the terminal reward function is supposed to be Borelian only and when it is continuous. We first establish a weak dynamic programming principle by using some refined results recently provided in \cite{DQS} and some properties of doubly reflected BSDEs with jumps (DRBSDEs). We then show a stronger dynamic programming principle in the continuous case, which cannot be derived from the weak one. In particular, we have to prove that the value function of the problem is continuous with respect to time $t$, which requires some technical tools of stochastic analysis and some new results on DRBSDEs. We finally study the links between our mixed problem and generalized Hamilton Jacobi Bellman variational inequalities in both cases.