A stochastic representation problem with applications in optimal singular control, Dynkin games, and obstacle problems

Singular control problems of bounded variation have been studied extensively via, dynamic programming which yields variational inequalities. In this thesis, we propose a new approach based on the problem of representing two given optional processes in terms of reflections of another two processes. As an application, we show how to solve a variant of two-sided Skorohod's obstacle problem in the context of backward stochastic differential equations.;The stochastic representation problem is of intrinsic mathematical interest. We start with defining reflection processes induced by two random processes. The strong connections between bounded-variation control and Dynkin games suggest us to involve an auxiliary family of Dynkin games to show the existence theorem. Two representation processes are then constructed from the value processes of the auxiliary family of Dynkin games. Local representation property is obtained from the construction of representation processes and saddle points of the family of Dynkin games. At last, we invoke Zorn's lemma to give a transfinite induction proof to extend the local representation to the global one.;A uniqueness theorem is presented. Up to indistinguishability, there can be at most one pair of representation processes to the stochastic representation problem. They are characterized as the value processes of two non-standard Dynkin games.