On Drift-Implicit and Full-Implicit Euler-Maruyama Methods for Solution of First Order Stochastic Differential Equations

Abstract

This paper examines the approximate solution of general first order stochastic differential equations (SDEs). Two different methods of solution were considered. They include Drift-implicit Euler-Maruyama method (DIEMM) and full-implicit Euler-Maruyama method (FIEMM). The two methods were adapted from Explicit Euler-Maruyama method (EEMM). Two problems in the form of first order SDEs considered are Black-Scholes option price model (BSOPM) with a drift and without a drift function. The absolute errors were calculated using the exact solution and numerical solution for stepsizes 2^-4, 2^-5, 2^-6, 2^-7, 2^-8, 2^-9.   Comparison in performance of the methods was achieved using mean absolute error criterion. The mean absolute error was then used to determine the order of convergence for each method. The order of convergence obtained for DIEMM and FIEMM was compared with that obtained using EEMM. The results showed that the performance of EEMM was better than DIEMM while the performance of FIEMM was better than that of EEMM and DIEMM for first problem. It was noted that the order of convergence of EEMM and DIEMM are approximately the same in the second problem. This can be associated with the absence of Drift function in second problem. However, FIEMM outperformed EEMM and DIEMM. This is because its order of convergence was less than that of EEMM and DIEMM for second problem. Also, the graphical solutions were constructed for each method can also be identified for stepsize 2^-4.