A weak trapezoidal method for a class of stochastic differential equations

TL;DR

This paper introduces a simple, second-order accurate numerical method for a specific class of stochastic differential equations driven by Brownian motions, useful in population and chemical kinetics modeling.

Contribution

The paper presents a new weak second-order numerical method that avoids complex calculations like iterated Ito integrals, simplifying the approximation process.

Findings

Constructs paths with second order weak accuracy.

Does not require derivatives or iterated Ito integrals.

Applicable to population and chemical reaction models.

Abstract

We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated Ito integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. This fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order.