Uniqueness conjecture for viscosity solutions of the mixed stochastic differential game

Let $V^+$ and $V^-$ be the upper and lower value functions of the stochastic differential game of mixed type, defined by

$$V^+(x)=\sup_{\hat{\beta}\in\hat{\Delta}}\inf_{u_1\in\mathcal{A}_1,\,\theta\geq 0}R(x,u_1(\cdot),\theta,\hat{\beta}[u_1](\cdot)),$$

and

$$V^-(x)=\inf_{\hat{\alpha}\in\hat{\Gamma}}\sup_{u_2\in\mathcal{A}_2,\,\tau\geq 0}R(x,\hat{\alpha}[u_2](\cdot),u_2(\cdot),\tau).$$

Here $\mathcal{A}_1$ and $\mathcal{A}_2$ are the admissible control sets, $\hat{\Gamma}$ and $\hat{\Delta}$ are the corresponding sets of non-anticipating control-and-stopping strategies, and

$$and$$

are the two viscosity-solution equations associated with the game. Uniqueness conjecture. The value functions $V^+$ and $V^-$ are unique viscosity solutions of

$$and$$

, respectively, in the class of bounded continuous functions. The conjecture is true for the special case treated in Section 3. Analogous results hold when the matrix $a$ is independent of the control variables and is uniformly elliptic, while the general statement remains unresolved.