Exact higher-order moments for linear non-homogeneous stochastic differential equation

Article type: Research Article

Authors: Guidoum, Arsalane Chouaib^{a; *} | Boukhetala, Kamal^b

Affiliations: [a] Department of Mathematics and Computer Science, Faculty of Sciences and Technology, University of Tamanghasset, Tamanghasset, Algeria | [b] Faculty of Mathematics, University of Science and Technology Houari Boumediene, Algeris, Algeria

Correspondence: [*] Corresponding author: Arsalane Chouaib Guidoum, University of Tamanghasset, Tamanghasset, 11000, Algeria. Fax: +213 29 30 00 71; E-mail: [email protected].

Abstract: This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Keywords: Higher-order moments, stochastic differential equation, time-dependent coefficients, Itô’s formula

DOI: 10.3233/MAS-231435

Journal: Model Assisted Statistics and Applications, vol. 18, no. 4, pp. 321-329, 2023