Applied Lyapunov Stability for Some Nonlinear Stochastic Differential Equations
Article Sidebar
Main Article Content
Abstract
In this paper, we applied and explain the stability to some linear and non-linear stochastic differential equations by using the Lyapunov direct second method, after finding the stochastic differential equation which obtained by applying the (Ito-integrated formula) and the quadratic Lyapunov function be taken, we use the Lyapunov theorems to find and explain if the trivial (zero) solution are stochastically stabile (p-stable, mean square stable and stochastically asymptotically stable in the large ), then we explain the methods by some examples.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Tikrit Journal of Pure Science is licensed under the Creative Commons Attribution 4.0 International License, which allows users to copy, create extracts, abstracts, and new works from the article, alter and revise the article, and make commercial use of the article (including reuse and/or resale of the article by commercial entities), provided the user gives appropriate credit (with a link to the formal publication through the relevant DOI), provides a link to the license, indicates if changes were made, and the licensor is not represented as endorsing the use made of the work. The authors hold the copyright for their published work on the Tikrit J. Pure Sci. website, while Tikrit J. Pure Sci. is responsible for appreciate citation of their work, which is released under CC-BY-4.0, enabling the unrestricted use, distribution, and reproduction of an article in any medium, provided that the original work is properly cited.
References
[1] Fischer, S., and R.C. Merton, "Macroeconomics and Finance; The Role of the Stock Market," Carnegie Rochester Conference Series on Public Policy. Vol. XXI, pp. 57-108. (1984).
[2] Jha, S.K. and Langmead, C.J. Exploring behaviors of stochastic differential equation models of biological systems using change of measures. BMC Bioinformatics 13 (Suppl 5), S8 (2012).
[3] Shaobo Zhou, "Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations", Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 127–14,2014.
[4] BILLUI KAYMAKALAN “STABILITY THEORY FOR DYNAMIC SYSTEMS ON TIME SCALES” INTERNATIONAL, Journal of Applied Mathematics and Stochastic Analysis, Vol.5, no. 3, pp. 275-282,1992.
[5] Hu, L., Mao, X., & Yi, S.” Stability and boundedness of nonlinear hybrid stochastic differential delay equations”. IEEE, 2013.
[6] Erkan Nane and Yinan Ni “Path stability of stochastic differential equations driven by time-changed L´evy noises” ALEA, Lat. Am. J. Probab. Math. Stat. 15, 479–507,2018.
[7] Ayman M. Elbaz, William L. Roberts “ Structure and stability characteristics of turbulent planar flames with inhomogeneous jet in a concentric flow slot burner” Proceedings of the Combustion Institute Volume 38, Issue 2,pp. 2597-2606, 2021.
[8] Dzhalladova I., Ruzickova M. and Ruzickova V., "Stability of The Zero Solution of Nonlinear Differential Equations Under the Influence of White Noise". Article in Advances in Difference Equations, A springer open journal, 2015.
[9] Rochambeau C., Corn ford D., et al., "Gaussian Process Approximations of Stochastic Differential Equations", JMLR: Workshop and Conference Proceedings, Vol.1, pp.1-16, 2007.
[10] Arnold L., "Stochastic Differential Equations; Theory and Applications", John Wiley and Sons, Inc., 1974.
[11] Dzhalladova I., Ruzickova M. and Ruzickova V., "Stability of The Zero Solution of Nonlinear Differential Equations Under the Influence of White Noise". Article in Advances in Difference Equations, A springer open journal, 2015.