An extension of stochastic differential models by using the Grunwald-Letnikov fractional derivative

Subject Areas : Applied Mathematics

Mohammad Ali Jafari ¹ , Narges Mousaviy ²

1 - Department of Financial Sciences, Kharazmi University, P.O. Box 15875-1111, Tehran, Iran.
2 - Department of Financial Sciences, Kharazmi University, P.O. Box 15875-1111, Tehran, Iran.

Received: 2019-12-31 Accepted : 2020-06-17 Published : 2022-05-01

Keywords: Stochastic process with memory, Stochastic differential model, Grunwald-Letnikov fractional derivative, fractional derivative,

Abstract :

Stochastic differential equations (SDEs) have been applied by engineers and economists because it can express the behavior of stochastic processes in compact expressions. In this paper, by using Grunwald-Letnikov fractional derivative, the stochastic differential model is improved. Two numerical examples are presented to show efficiency of the proposed model. A numerical optimization approach based on least square approximation is applied to determine the order of the fractional derivative. Numerical examples show that the proposed model works better than the SDE to model stochastic processes with memory.