On the stochastic control-stopping problem

We study the stochastic control-stopping problem when data are path-dependent and of polynomial growth. The approach is based on backward differential equations (BSDEs for short). turns into a specific reflected BSDE with Lipschitz coefficient which we show existence uniqueness solution. then establish its relationship value function problem. optimal strategy exhibited. Finally in Markovian framework prove that unique viscosity solution associated Hamilton-Jacobi-Bellman equation.