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

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.

💡 Research Summary

The paper studies a zero‑sum Dynkin game driven by a stochastic differential equation (SDE) whose coefficients are random and depend on the path of an auxiliary Brownian motion (B). The state process (X) is therefore non‑Markovian, although the coefficients are (\mathcal{F}^B)-predictable, which guarantees a uniform parabolic condition. The authors first extend the Krylov estimate, originally developed for Markov processes, to this non‑Markov setting by employing the maximum principle for quasilinear backward stochastic partial differential equations (BSPDEs) due to Qiu and Tang. With this estimate and a smoothing argument they derive a generalized Itô‑Kunita‑Wentzell formula that allows test functions belonging to a suitable Sobolev space and being random fields of Itô type. This formula is crucial for evaluating (V_t(X_t)) when the candidate value function (V) is only a strong solution (i.e., its second spatial derivatives exist in (L^2) but are not necessarily continuous).

The payoff of the Dynkin game consists of a running cost (f) and two obstacle functions (\underline V) (lower) and (\overline V) (upper). For any pair of stopping times ((\tau_1,\tau_2)) the payoff functional is defined in the usual way, and a Nash equilibrium ((\tau_1^,\tau_2^)) is sought. The main verification theorem shows that the game value (V_t(x)=\mathbb{E}