Boundary sensitivities for di usion processes in time dependent domains

We study the sensitivity, with respect to a time dependent domain Ds, of expectations of functionals of a di usion process stopped at the exit from Ds or normally re ected at the boundary of Ds. We establish a di erentiability 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.