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| √ | ln |z||) in the state variables y and z. The terminal value ξ and the obstacle processes (Lt)0≤t≤T and (Ut)0≤t≤T are L-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.