Combinatorics for calculating expectations of stochastic differential equations

Combinatorial discussion is proposed and applied for calculating expectations of stochastic differential equations. Starting from the duality theory of stochastic processes, some modifications of interpretation and usages of time-ordering operators naturally lead to combinatorial discussions. As a demonstration, the first and second moments for the Ornstein-Uhlenbeck process are re-derived from the combinatorial discussion. Furthermore, two numerical methods for practical applications are proposed. One is based on a conventional exponential expansion and the Pade approximation. Another uses a resolvent of a time-evolution operator, and the Aitken series acceleration method is also employed. These two proposals recover the correct results approximately.