Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

Volume 33, Issue 4, pp 408--419 https://dx.doi.org/10.22436/jmcs.033.04.07

Publication Date: January 31, 2024 Submission Date: December 15, 2023 Revision Date: January 02, 2024 Accteptance Date: January 04, 2024

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Authors

A. M. A. El-Sayed - Faculty of Science, Alexandria University, Egypt. M. E. I. El-Gendy - Department of Mathematics, College of Science and Arts at Al-Nabhaniah, AL Qassim University, Al-Nabhaniah, Kingdom of Saudi Arabia. - Department of Mathematics, Faculty of Science, Damanhour University, Egypt.

Abstract

Stochastic problems play a huge role in many applications including biology, chemistry, physics, economics, finance, mechanics, and several areas. In this paper, we are concerned with the nonlocal stochastic-integral problem of the arbitrary (fractional) orders stochastic differential equation \[ \frac{dX(t)}{dt}=f_{1}(t,D^{\alpha} X(t))+f_{2}(t,B(t)), \ \ t\in(0,T], \qquad X(0)=X_0 +\int_0^Tf_3(s, D^{\beta} X(s)) dW(s), \] where \(B\) is any Brownian motion, \(W\) is a standard Brownian motion, and \(X_0\) is a second order random variable. The Hyers - Ulam stability of the problem will be studied. The existence of solution and its continuous dependence on the Brownian motion \(B\) will be proved. The three spatial cases Brownian bridge process, the Brownian motion with drift and the Brownian motion started at \(A\) will be considered.

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Keywords

• Stochastic processes
• stochastic differential equations
• existence of solutions
• continuous dependence
• Brownian motion
• Brownian bridge process
• Brownian motion with drift

MSC

•  60H20
•  26A33
•  45K05

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