Optimal Switching of One-Dimensional Reflected BSDEs and Associated Multidimensional BSDEs with Oblique Reflection

In this paper, the optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (BSDEs, for short) where the generators, the terminal values, and the barriers are all switched with positive costs. The value process is characterized by a system of multi-dimensional reflected BSDEs with oblique reflection, whose existence and uniqueness is by no means trivial and is therefore carefully examined. Existence is shown using both methods of the Picard iteration and penalization, but under some different conditions. Uniqueness is proved by representation either as the equilibrium value process to a stochastic mixed game of switching and stopping, or as the value process to our optimal switching problem for one-dimensional reflected BSDEs.