Before revision: In this paper we consider the following optimal stopping problem V^{\omega}_{ \rm A }(s) = \tau\in\mathcal{T} E_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)], where the process S_t is a jump-diffusion process, T is a family of stopping times and g and \omega are fixed payoff function and discounting function, respectively.
After revision: In this paper we consider the following optimal stopping problem V^{\omega}_{ \rm A }(s) = \tau\in\mathcal{T} E_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)], where the process S_t is a jump-diffusion process, T is a family of stopping times and g and \omega are fixed payoff function and discount function, respectively.