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Stochastic differential equations are widely used in various fields; inparticular, the usefulness of duality relations has been demonstrated in somemodels such as population models and Brownian momentum processes. In thisstudy, a discussion based on combinatorics is made and applied to calculate theexpectation values of functions in systems in which evolution is governed bystochastic differential equations. Starting with the duality theory ofstochastic processes, some modifications to the interpretation and usage oftime-ordering operators naturally lead to discussions on combinatorics. Fordemonstration, the first and second moments of the Ornstein-Uhlenbeck processare re-derived from the discussion on combinatorics. Furthermore, two numericalmethods for practical applications are proposed. One method is based on aconventional exponential expansion and the Pade approximation. The other uses aresolvent of a time-evolution operator, along with the application of theAitken series acceleration method. Both methods yield reasonableapproximations. Particularly, the resolvent and Aitken acceleration showsatisfactory results. These findings will provide a new way of calculatingexpectations numerically and directly without using time-discretization.

Combinatorics for calculating expectation values of functions in systems with evolution governed by stochastic differential equations