Optimal stopping under g-Expectation with -integrable reward process

Abstract

AbstractIn this paper we study a class of optimal stopping problems under g-expectation, that is, the cost function is described by the solution of backward stochastic differential equations (BSDEs). Primarily, we assume that the reward process is $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable with $\mu>\mu_0$ for some critical value $\mu_0$ . This integrability is weaker than $L^p$ -integrability for any $p>1$ , so it covers a comparatively wide class of optimal stopping problems. To reach our goal, we introduce a class of reflected backward stochastic differential equations (RBSDEs) with $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable parameters. We prove the existence, uniqueness, and comparison theorem for these RBSDEs under Lipschitz-type assumptions on the coefficients. This allows us to characterize the value function of our optimal stopping problem as the unique solution of such RBSDEs.