Comparative study of Stochastic Taylor Methods and Derivative-Free Methods for Stochastic Differential Equations

Abstract

Ordinary differential equations (ODEs) have been widely used to model the dynamical behaviour of biological and physical systems. However, modelling these systems using deterministic model such as ODEs is inadequate as the system is subjected to the uncontrolled factors of environmental noise. Stochastic differential equations (SDEs) which are originating from the irregular Brownian motion can be applied to model such systems that subjected to the uncontrolled factors of noisy behaviour. Numerical methods are required to approximate the solution of the model due to the complexity of the equation. Theoretically, methods with a higher order of convergence lead to a better approximation of the solutions. Though, the implementation of the methods might not reflect the theoretical finding as generating a lot of random numbers might contribute to the instability of the methods. This research is aimed to investigate the performance of stochastic Taylor methods of Euler-Maruyama, Milstein scheme, and derivative-free methods of second and fourth-order stochastic Runge-Kutta in approximating the solution of SDEs. Four types of mathematical models which include the Black-Scholes model, logistic model, stochastic Gompertz model, and prey-predator model are simulated using the aforementioned numerical methods. Numerical solutions of the Black-Scholes model are compared with the analytical solution meanwhile, the numerical solutions of the logistic model and stochastic Gompertz models are compared with the experimental data of the fermentation process and cancer cell growth, respectively. The simulated results of the prey-predator model are compared with the experimental data of the interaction between cancer cells (prey) and anticancer Chrondoitin Sulfate (predator). The prediction performance of the methods is measured using global error and root mean square errors (RMSE).