Comparison Between Milstein and Exact Coupling Methods using MATLAB for a Particular Two-Dimensional Stochastic Differential Equation

We compare Milstein and exact coupling methods for the strong approximation of solutions to stochastic differential equations (SDE), which are driven by Brownian motion. Both of these methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the exact coupling method. We also compare their implementation using MATLAB. A particular two-dimensional SDE is used in the implementation for comparing their results. Moreover, the performance of both methods and the amount of time required to obtain the result are also analyzed. It is interesting to mention that this comparison is very important in several areas, such as stochastic analysis, financial mathematics and some physical applications.