Exact Calculation of Stochastic Time Integrals with an Application Towards High Order Monte Carlo of Non-Linear Stochastic Differential Equations

Abstract

Drift, variance and other time integrals of functions of a stochastic process are again stochastic processes and their stochastic integrals admit expansions of higher order in the time step used for the discretization of the stochastic differential equations. We show how to derive high order and almost exact expansions of time integrals of functions of a stochastic process using Ito change of variable formula. We explain those ideas in some detail in this paper and show how those ideas could be used for high-order, robust and precision monte carlo of stochastic differential equations where traditional monte carlo always failed. Current monte carlo practice does not properly take into account the stochastic nature of evolution of mean and variance of a stochastic process and traditional monte carlo mimicks the true evolution of a stochastic differential equation only when the SDE is “close to linear.” In monte carlo, we always take a finite time step, which makes parts of SDEs as stochastic time integrals but we use the equation for “infinitesimal dt” which is not valid for larger time intervals. We show how to accurately calculate these stochastic time integrals for large steps even when the stochastic differential equation is highly non-linear.