Statistics for stochastic differential equations and local approximation of Crank-Nicolson method

Numerical evaluation of statistics plays an important role in data assimilation and filtering. When one focuses on stochastic differential equations, Monte Carlo simulations or moment closure approximations are available to evaluate the statistics. The other approach is to solve the corresponding backward Kolmogorov equation. However, a basis expansion for the backward Kolmogorov equation leads to infinite systems of differential equations, and conventional numerical methods such as the Crank-Nicolson method are not available directly. Here, a local approximation of the Crank-Nicolson method is proposed. The local approximation transforms the implicit time integration algorithm into an explicit one, which enables us to employ combinatorics for the proposed algorithm. The proposed algorithm shows a second-order convergence. Furthermore, the convergence property naturally leads to extrapolation methods; they work well to calculate a more accurate value with small steps. The proposed method is demonstrated with the Ornstein-Uhlenbeck process and the noisy van der Pol system.