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
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.
Share and Cite
ISRP Style
A. M. A. El-Sayed, M. E. I. El-Gendy, Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation, Journal of Mathematics and Computer Science, 33 (2024), no. 4, 408--419
AMA Style
El-Sayed A. M. A., El-Gendy M. E. I., Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation. J Math Comput SCI-JM. (2024); 33(4):408--419
Chicago/Turabian Style
El-Sayed, A. M. A., El-Gendy, M. E. I.. "Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation." Journal of Mathematics and Computer Science, 33, no. 4 (2024): 408--419
Keywords
- Stochastic processes
- stochastic differential equations
- existence of solutions
- continuous dependence
- Brownian motion
- Brownian bridge process
- Brownian motion with drift
MSC
References
-
[1]
Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation, Mixed Caputo fractional neutral stochastic differential equations with impulses and variable delay, Fractal Fract., 5 (2021), 1–19
-
[2]
S. R. Aderyani, R. Saadati, T. M. Rassias, H. M. Srivastava, Existence, uniqueness and the multi-stability results for a -Hilfer fractional differential equation, Axioms, 12 (2023), 16 pages
-
[3]
R. P. Agarwal, B. Xu, W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852–869
-
[4]
E. Ahmed, A. M. A. El-Sayed, A. E. M. El-Mesiry, H. A. A. El-Saka, Numerical solution for the fractional replicator equation, Int. J. Mod. Phys. C, 16 (2005), 1017–1025
-
[5]
M. Ahmad, A. Zada, M. Ghaderi, R. George, S. Rezapour, On the existence and stability of a neutral stochastic fractional differential system, Fractal Fract., 6 (2022), 1–16
-
[6]
A. Babaei, H. Jafari, S. Banihashemi, A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise, Symmetry, 12 (2020), 15 pages
-
[7]
R. F. Curtain, A. J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, London-New York (1977)
-
[8]
M. M. El-Borai, On some stochastic fractional integro-differential equations, Adv. Dyn. Syst. Appl., 1 (2006), 49–57
-
[9]
M. M. Elborai, A. A. Abdou, W. El-Sayed, S. I. Awed, Numerical methods for solving integro partial differential equation with fractional order, J. Posit. Sch. Psychol., 6 (2022), 2124–2134
-
[10]
M. M. Elborai, K. E. S. El-Nadi, Stochastic fractional models of the diffusion of COVID-19, Adv. Math. Sci. J., 9 (2020), 10267–10280
-
[11]
A. M. A. El-Sayed, On stochastic fractional calculus operators, J. Fract. Calc. Appl., 6 (2015), 101–109
-
[12]
A. M. A. El-Sayed, A. Arafa, A. Haggag, Mathematical Models for the Novel coronavirus (2019-nCOV) with clinical data using fractional operator, Numer. Methods Partial Differential Equations, 39 (2023), 1008–1029
-
[13]
A. M. A. El-Sayed, F. Gaafar, M. El-Gendy, Continuous dependence of the solution of random fractional order differential equation with nonlocal condition, Fract. Differ. Calc., 7 (2017), 135–149
-
[14]
A. M. A. El-Sayed, M. E. I. El-Gendy, Solvability of a stochastic differential equation with nonlocal and integral condition, Differ. Uravn. Protsessy Upr., 2017 (2017), 47–59
-
[15]
A. M. A. El-Sayed, H. A. Fouad, On a coupled system of stochastic Ito differential and the arbitrary (fractional)order differential equations with nonlocal random and stochastic integral conditions, Mathematics, 9 (2021), 14 pages
-
[16]
A. M. A. El-Sayed, H. A. Fouad, On a coupled system of random and stochastic nonlinear differential equations with coupled nonlocal random and stochastic nonlinear integral conditions, Mathematics, 9 (2021), 13 pages
-
[17]
A. M. A. El-Sayed, F. M. Gaafar, Fractional calculus and some intermediate physical processes, Appl. Math. Comput., 144 (2003), 117–126
-
[18]
A. M. A. El-Sayed, F. Gaafar, M. El-Gendy, Continuous dependence of the solution of Ito stochastic differential equation with nonlocal conditions, Appl. Math. Sci., 10 (2016), 1971–1982
-
[19]
A. M. A. El-Sayed, H. H. G. Hashim, Stochastic Itˆo-differential and integral of fractional-orders, J. Fract. Calc. Appl., 13 (2022), 251–258
-
[20]
F. M. Hafez, The fractional calculus for some stochastic processes, Stochastic Anal. Appl., 22 (2004), 507–523
-
[21]
M. A. Hamdy, M. M. El-Borai, M. E. Ramadan, Noninstantaneous impulsive and nonlocal Hilfer fractional stochastic integro-differential equations with fractional Brownian motion and Poisson jumps, Int. J. Nonlinear Sci. Numer. Simul., 22 (2021), 927–942
-
[22]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222–224
-
[23]
D. H. Hyers, The stability of homomorphisms and related topics, Teubner, Leipzig, 57 (1983), 140–153
-
[24]
H. Jafari, D. Uma, S. R. Balachandar, S. G. Venkatesh, A numerical solution for a stochastic beam equation exhibiting purely viscous behavior, Heat Transf., 52 (2023), 2538–2558
-
[25]
B. Kafash, R. Lalehzari, A. Delavarkhalafi, E. Mahmoudi, Application of stochastic differential system in chemical reactions via simulation, MATCH Commum. Math. Comput. Chem., 71 (2014), 265–277
-
[26]
O. Knill, Probability Theory and Stochastic Process with Applications, Narinder Kumar Lijhara for Overseas Press India Private Limited, India ()
-
[27]
Q. Li, Y. Zhou, The existence of mild solutions for Hilfer fractional stochastic evolution equations with order 2 (1, 2), Fractal Fract., 7 (2023), 1–23
-
[28]
M. Medved’, M. Posp´ıˇsil, E. Brestovansk´, Nonlinear integral inequalities involving tempered -Hilfer fractional integral and fractional equations with tempered -Caputo fractional derivative, Fractal Fract., 7 (2023), 1–17
-
[29]
B. Oksendal, Stochastic differential equations: an introduction with applications, Springer-Verlag, Heidelberg, New York (2000)
-
[30]
O. Posch, Advanced Macroeconomics, University of Hamburg, (2010)
-
[31]
S. Z. Rida, A. M. A. El-Sayed, A. A. M. Arafa, Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model, J. Stat. Phys., 140 (2010), 797–811
-
[32]
T. T. Soong, Random differential equations in science and engineering, Academic Press, New York-London (1973)
-
[33]
D. Uma, H. Jafari, S. R. Balachandar, S. G. Venkatesh, A mathematical modeling and numerical study for stochastic Fisher-SI model driven by space uniform white noise, Math. Methods Appl. Sci., 46 (2023), 10886–10902
-
[34]
B.-H. Wang, Y.-Y. Wang, C.-Q. Dai, Y.-X. Chen, Dynamical characteristic of analytical fractional solutions for the spacetime fractional Fokas-Lenells equation, Alex. Eng. J., 59 (2020), 4699–4707
-
[35]
E. Wong, Stochastic Processes, Informations and Dynamical Systems, McGraw-Hill, USA (1971)
-
[36]
E. Wong, Introduction to random processes, Springer-Verlag, New York (1983)