Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations.

Abstract

In this article we suggest a new method for solutions of stochastic integrals where the dynamics of the variables in integrand are given by some stochastic differential equation. We also propose numerical simulation of stochastic differential equations which is based on iterated integrals method employing Ito change of variable expansion of the drift and volatility function of the SDE. If a function has to be evaluated along time, it would be possible to calculate its value in time using the initial value of the function and an integral of the time derivative of the function till terminal time. Method of iterated integrals is a method of solution of integrals in which we repeatedly expand different time dependent terms in the integral as an initial value and integral of its derivative along time so that their sum would equal the original term in the integrand that was expanded. We continue to repeatedly expand the integrand in time dependent terms into initial value terms and integral of further time dependent higher derivative and continue until we have enough accuracy in the terms evaluated at initial values so as to be able to neglect the time dependent highest derivative terms. We evaluate all these iterated integrals terms at the initial time of the simulation along with the solution of the stochastic integrals which is found in terms of Hermite polynomials and variance of the integrals. We apply the method of iterated integrals to simulation of various stochastic differential equations. We find that the accuracy of simulation sharply increases using the method of iterated integral as compared to naive and simple monte carlo simulation methods.