BSDE Approach for Dynkin Game and American Game Option

Abstract

Consider a Dynkin game with payoff

$$ J(\lambda , {\upsigma }) = F\bigg [U_{\lambda }1_{\{\lambda< {\upsigma }\}} + L_{{\upsigma }}1_{\{\lambda > {\upsigma }\}}+ Q_{{\upsigma }}1_{\{ {\upsigma }=\lambda < T \}} + \xi 1_{\{ {\upsigma }= \lambda = T \}}\bigg ], $$

where $F : \mathbb {R}\longrightarrow \mathbb {R}$ is a continuous nondecreasing function and $\lambda , {\upsigma }$ are stopping times valued in [0, T]. We show the existence of a value as well as a saddle-point for this game using the theory of BSDE with double reflecting barriers. An American game option pricing problem is also discussed.