Boundary sensitivities for diffusion processes in time dependent domains

We study the sensitivity, with respect to a time dependent domain \D_s, of expectations of functionals of a diffusion process stopped at the exit from \D_s or normally reflected at the boundary of \D_s. We establish a differentiability result and give an explicit expression for the gradient that allows the gradient to be computed by Monte Carlo methods. Applications to optimal stopping problems and pricing of American options, to singular stochastic control and others are discussed.