Numerical approaches to stochastic differential equations with boundary conditions

We consider stochastic differential equations in convex domains in $R\sp{d}$ with reflection at the boundary. (The solutions are known as reflected diffusions.) We develop three strong schemes to approximate the solution numerically, and obtain the rate of convergence in the mean square sense. We then modify the results and propose two weak schemes to approximate the expectations of functionals of the solutions and we obtain weak convergence of order one; here we use the diffusion property by exploiting relationships given by the infinitesimal generator (a PDE operator with Neumann boundary conditions) of the diffusion. Next we propose a Monte-Carlo method to solve numerically certain elliptic PDEs in convex domains with Neumann boundary conditions. The cost is less than that of the finite difference method for these PDEs when the state space dimension is large, since the number of calculations is exponentially dependent on the dimension for the finite difference method while it is only linearly dependent on the dimension for the Monte-Carlo method. We then show a few examples on how to formulate some management problems as stochastic differential equations with boundary conditions. Finally we simulate on a computer the solutions of some reflected diffusions using our schemes.