Comparison of Euler scheme and Milstein scheme for stochastic differential equations

Abstract

The solutions of stochastic differential equations(SDEs) are given by stochastic processes. These are two well-known numerical schemes for solving SDEs, the Euler scheme and the Milstein scheme. They are based on Ito integrals and stochastic Taylor expansions. Assuming that stochastic numerical solutions converge to theoretical solution, we define the strong convergence type and say that the stochastic numerical schemes converge to solution with strong order $\gamma$. The Euler scheme converges with strong order 0.5 and the Milstein scheme converges with strong order 1. In general, asset price model which is following SDE, $dX_t = \mu X_tdt + σ X_tdW_t$, is well-known in financial mathematics and we can find the analytic solution of SDE following asset price model by using Ito formula. In practice, we compute the expectation of errors between the numerical solutions and theoretical solution, the 95%-confidence intervals of errors and the strong orders of asset price model. We can compare the accuracy of Euler scheme and Milstein scheme. Using two kinds of stochastic numerical schemes we can simulate the solutions of SDEs.