Combinatorics for calculating expectations of stochastic differential equations

Jun Ohkubo

Published 2020-05-09Version 1

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.