Measure order of convergence without an exact solution, Euler vs Milstein scheme

Abstract

The purpose of this paper is to measure the strong and weak order of convergence of both the Euler and Milstein schemes using a stochastic volatility model and an N−dimensional Exponential Brownian Motion Process (EBM). An exact solution is normally required to calculate the order of convergence, however there are none available for this volatility process. We propose a method to solve this problem. We also show numerically that when we apply the Milstein scheme to an N−dimensional stochastic process, there is a need to take into account the correlation between the systems.