Second order stochastic differential equations with Dirichlet boundary conditions

AbstractWe consider the second order stochastic differential equation Ẍt + 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(1) = b. We show pathwise existence and uniqueness of a solution assuming some smoothness and monotonicity conditions on f, and we study the Markov property of the solution using an extended version of the Girsanov theorem due to Kusuoka