Solution of the first boundary value problem for parabolic equations by integration of stochastic differential equations

The solution of the general Dirichlet problem is connected in a well-known manner with a system of stochastic differential equations. The paper describes a number of methods of constructing a Markovian chain with absorbtion weakly approximating the system solutions so that the expectation of a certain functional in the chain trajectories is close to the solution of the original problem. For the examined methods the convergence theorems are established indicating the order of accuracy with respect to the size of approximation step.