Numerical calculation of E[f(X(1,x))] where {X(t,x)}_t≧0 is a diffusion process defined by a stochastic differential equation (SDE) is called weak approximation of the SDE. The author discusses higherorder discretization algorithms based on Kusuokaʼs theory. The fundamental theorem by Kusuoka and three algorithms are explained.

Optical tomography is a new medical imaging technique which makes use of near infrared light, and it is mathematically modeled as an inverse problem to determine a coefficient in the stationary transport equation from boundary measurements. In this paper, we show two results on regularity of solutions to the direct problem of the stationary transport equation. The first result is to describe discontinuity of the solution which arises from discontinuous points of the incoming boundary data, and we show the exponential decay of a jump of the solution on a discontinuous point. The decay gives an idea for solving the inverse problem. The second one is to give a W^{1, p} estimate of the solution to the direct problem in two-dimensional cases. Here, the exponent p has an upper bound p_m, which is a real number depending only on the shape of the domain. This estimate is crucial when we solve the direct problem numerically with the discrete-ordinate discon-tinuous Galerkin method.