Solving stochastic differential equations via ordinary differential equations

Abstract

Stochastic differential equations (SDEs) are a useful model for quantities that vary over time. To study its properties, we often have to efficiently solve an SDE numerically. The classical way to obtain high-order strong solution schemes is via the Itô-Taylor expansion, but this can be inconvenient in practice. We apply the Wong-Zakai theorem to convert the SDE into an ODE and then solve this ODE numerically. We present two methods to make this work in practice. The first method aims to largely solve the SDE in question without knowledge of the specific Brownian path, but rests on an assumption which holds true only for a subset of SDEs. The second method approximates the Brownian path with moment matching and then solves the resulting ODE with Runge-Kutta 4; for a piecewise linear approximation this method appears promising and works regardless of the exact analytical expression of the SDE.