  Stochastic differential equations are widely used in various fields; in
particular, the usefulness of duality relations has been demonstrated in some
models such as population models and Brownian momentum processes. In this
study, a discussion based on combinatorics is made and applied to calculate the
expectation values of functions in systems in which evolution is governed by
stochastic differential equations. Starting with the duality theory of
stochastic processes, some modifications to the interpretation and usage of
time-ordering operators naturally lead to discussions on combinatorics. For
demonstration, the first and second moments of the Ornstein-Uhlenbeck process
are re-derived from the discussion on combinatorics. Furthermore, two numerical
methods for practical applications are proposed. One method is based on a
conventional exponential expansion and the Pade approximation. The other uses a
resolvent of a time-evolution operator, along with the application of the
Aitken series acceleration method. Both methods yield reasonable
approximations. Particularly, the resolvent and Aitken acceleration show
satisfactory results. These findings will provide a new way of calculating
expectations numerically and directly without using time-discretization.
