Optimal Stopping Problem for Stochastic Differential Equations with Random Coefficients

An optimal stopping problem for stochastic differential equations with random coefficients is considered. The dynamic programming principle leads to a Hamiltion–Jacobi–Bellman equation, which, for the current case, is a backward stochastic partial differential variational inequality (BSPDVI, for short) for the value function. Well-posedness of such a BSPDVI is established, and a verification theorem is proved.