On the variance of the estimate of the functional of the diffusion process in a domain with a reflecting boundary

The estimation of the functional of the diffusion process in a domain with a reflecting boundary, which is obtained on the basis of numerical modeling of its trajectories, is considered. The value of this functional coincides with the solution at a given point of a boundary value problem of the third kind for a parabolic equation. A formula is obtained for the limiting value of the variance of this estimate under decreasing step in the Euler method. To reduce the variance of the estimate, a transformation of the boundary value problem is used, similar to the one that was previously proposed in the case of an absorbing boundary.