On the solutions set of non-local Hilfer fractional orders of an Itô stochastic differential equation
Authors
M. E. I. El-Gendy
- Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Burayda 51452, Saudi Arabia.
- Department of Mathematics, Faculty of Science, Damanhour University, Egypt.
Abstract
In this paper, an issue of Hilfer fractional order of an Itô stochastic differential equation with two non-local conditions is considered, studying case is split into two problems, one of them gives its solution as a second-order stochastic process and the other gives its solution as a non-standard Brownian motion in the same space of continuous second order processes. The existence of the solutions of both problems will be studied, the maximal and minimal solution will be defined, the sufficient conditions for uniqueness and some continuous dependencies will be shown. For some examples of non-standard Brownian motion as a Brownian motion with drift, geometric Brownian motion, Brownian bridge, and integrated Brownian bridge see [M.-Y. Chen, Department of Finance National Chung Hsing University, (2013), 1--19], [B. Kafash, R. Lalehzari, A. Delavarkhalafi, E. Mahmoudi, MATCH Commum. Math. Comput. Chem., \(\bf 71\) (2014), 265--277], [B. Øksendal, Springer-Verlag, Heidelberg New York, (2000)] and [O. Posch, University of Hamburg, (2010)].
Share and Cite
ISRP Style
M. E. I. El-Gendy, On the solutions set of non-local Hilfer fractional orders of an Itô stochastic differential equation, Journal of Mathematics and Computer Science, 35 (2024), no. 2, 149--168
AMA Style
El-Gendy M. E. I., On the solutions set of non-local Hilfer fractional orders of an Itô stochastic differential equation. J Math Comput SCI-JM. (2024); 35(2):149--168
Chicago/Turabian Style
El-Gendy, M. E. I.. "On the solutions set of non-local Hilfer fractional orders of an Itô stochastic differential equation." Journal of Mathematics and Computer Science, 35, no. 2 (2024): 149--168
Keywords
- Itô integral
- standard Brownian motion
- non-standard Brownian motion
- stochastically decreasing function
- stochastically increasing function
- stochastic maximal solution
- stochastic minimal solution
- fractional derivatives
- nonlinear equations
MSC
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