One dimensional reflected BSDEs with two barriers under logarithmic growth and applications

Abstract

In this paper we deal with the problem of the existence and the uniqueness of a solution for one dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator is allowing a logarithmic growth $(|y||\ln |y||+|z|\sqrt{|\ln |z||})$ in the state variables $y$ and $z$. The terminal value $\xi$ and the obstacle processes $(L_t{)}_{0\le t\le T}$ and $(U_t{)}_{0\le t\le T}$ are $L^p$-integrable for a suitable $p>2$. The main idea is to use the concept of local solution to construct the global one. As applications, we broaden the class of functions for which mixed zero-sum stochastic differential games admit an optimal strategy and the related double obstacle partial differential equation problem has a unique viscosity solution.

26 pages