  The Koopman operator is beneficial for analyzing nonlinear and stochastic
dynamics; it is linear but infinite-dimensional, and it governs the evolution
of observables. The extended dynamic mode decomposition (EDMD) is one of the
famous methods in the Koopman operator approach. The EDMD employs a data set of
snapshot pairs and a specific dictionary to evaluate an approximation for the
Koopman operator, i.e., the Koopman matrix. In this study, we focus on
stochastic differential equations, and a method to obtain the Koopman matrix is
proposed. The proposed method does not need any data set, which employs the
original system equations to evaluate some of the targeted elements of the
Koopman matrix. The proposed method comprises combinatorics, an approximation
of the resolvent, and extrapolations. Comparisons with the EDMD are performed
for a noisy van der Pol system. The proposed method yields reasonable results
even in cases wherein the EDMD exhibits a slow convergence behavior.
