Abstract:

We deal with the problem of existence and uniqueness of a solution for one-dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator has logarithmic growth \(|y|\,|\!\ln|y||+|z|\sqrt{|\!\ln|z||}\) in the state variables \(y\) and \(z\). The terminal value \(\xi\) and the obstacle processes \((L_t)_{0\leq t\leq T}\) and \((U_t)_{0\leq t\leq T}\) are \(L^p\)-integrable for a suitable \(p > 2\). The main idea is to use the concept of local solution to construct a 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.

2010 AMS Mathematics Subject Classification: Primary 91A60; Secondary 91A15, 60H10, 60H30.

Keywords and phrases: reflected BSDEs, mixed zero-sum stochastic differential game, penalization, viscosity solution.