Numerical Stabilities of Vasicek and Geometric Brownian Motion Models

Abstract

Stochastic differential equations (SDEs) are very often used as models for a large number of phenomena in the physical, economic and management sciences. They generalize the notion of ordinary differential equations, taking into account a white additive and multiplicative noise term, to model random trajectories such as stock market prices or particles movements, on the quantum scale, subject to diffusion phenomena. In rare cases, it is generally impossible to have explicit solution to these equations. In this case, the numerical approach, presenting itself under various aspects, is the only favorable outcome. However, the stability of numerical schemes for stochastic differential equations solution is much more significant. In this paper, we establish and make a classical proof of the mean and mean-square stabilities of the numerical SDEs schemes for Vasicek and Geometric Brownian motion models.