Stochastic mesh method for optimal stopping problems

The stochastic mesh method for solving a multidimensional optimal stopping problem for a diffusion process with non-linear payoff is considered. To solve the problem in the case of payoff for an Asian option with geometric average we provide a special discretization scheme for the diffusion process. This sampling scheme allows one to get rid of singularities in transition probabilities. Then, we consider transition probabilities of a stochastic mesh defined as averaged densities. Two estimates of the solution to the problem by the stochastic mesh method are given. The consistency of the defined estimates is proved. It is shown that the variance of the solution estimates is inversely proportional to the number of points in each mesh layer. The result extends the application area of the stochastic mesh method. A numerical example of the result is presented. We applying estimates to the call and put options compared to the option prices obtained through the regular mesh.