Second order stochastic differential equations with Dirichlet boundary conditions

Abstract

We consider the second order stochastic differential equation Xt + f(Xt, Xt) = Wt where t runs on the interval [0, 1], {Wt} is an ordinary Brownian motion and we impose the Dirichlet boundary conditions X(0) = a and X(l) = b. We show pathwise existence and uniqueness of a solution assuming sorne smoothness and monotonicity conditions on f, and we study the Markov property of the solution using an extended version of the Girsanov theorem dueto Kusuoka.