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 $(L_t{)}_{0\le t\le T}$ 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 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.