Approximate analytical solutions of linear stochastic differential models based on Karhunen-Loéve expansion with finite series terms

Abstract

Stochastic Differential Equations (SDEs) as particular forms of Differential Equations (DEs) play immense roles in modeling of various phenomena with applications in physical sciences, and finance- such as stock option practices due to thermal and random fluctuations. The solutions of these SDEs, if they exist, are difficult to obtain, unlike those of the Differential Equations. In this paper, the white noise terms of the linear SDEs in Stratonovich forms are considered on the basis of Karhunen-Loeve Expansion finite series while Daftardar-Jafari Integral Method is proposed for approximate analytical solution of the linear Stratonovich Stochastic Differential Equations. Three numerical examples are considered to test the accuracy and effectiveness of this proposed method. The results obtained show clearly that the approximate solutions converge faster to the exact solutions even with fewer terms; though, higher terms increase the accuracy. The method is direct in terms of application. Thus, it is recommended for nonlinear financial models such as Ito Stochastic Differential Equations.