Optimal stopping and backward stochastic differential equations

In this work we provide an alternative approach, based on Backward Stochastic Differential Equations, to optimal stopping theory. To this purpose, we study a particular class of Backward stochastic differential equations with jumps and a sign constraint. We prove the existence of a minimal solution by approximation via penalization. We then prove that the solution to such an equation provides an original representation for the value function of an optimal stopping problem, in the general context of non-Markovian stochastic processes. Moreover, we compare our solutions with the solutions to Reflected Backward stochastic differential equations, thus proving an explicit relationship between the former and the latter. Reflected equations are characterized by a ’reflection’ constraint, which keeps the solution above a given stochastic process, and provides a classical representation for the value function of an optimal stopping problem. Therefore, our result gives an equivalence between two representations for the value function of the optimal stopping problem, thus proving a new Backward stochastic differential equations approach to optimal stopping theory, based on equations with a sign constraint on the martingale component. At the end, in a Markovian framework, we show that, as a corollary of already known results, our method provides an alternative probabilistic representation for the unique viscosity solution to an obstacle problem for parabolic partial differential equations.