scholarly journals Bifurcation of a class of stochastic delay differential equations

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1821-1834
Author(s):  
Chaoliang Luo ◽  
Shangjiang Guo ◽  
Aiyu Hou
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Planck Equation ◽  
Original System ◽  
Stochastic Averaging ◽  
Small Time ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Polar Coordinate ◽  
Delay Differential

In this paper, we study the bifurcation of a class of two-dimensional stochastic delay differential equations. Firstly, we translate the original system into an It? limiting diffusion system by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then we discuss the dynamical bifurcation by analyzing the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. The obtained conclusions are completely new, which generalize and improve some existing results.

scholarly journals Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Peiguang Wang ◽  
Yan Xu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Delay Differential Equations ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Differential

In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H∈1/2,1. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.

scholarly journals A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yanli Zhou ◽  
Yonghong Wu ◽  
Xiangyu Ge ◽  
B. Wiwatanapataphee
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximation Scheme ◽  
Jump Diffusion ◽  
Mathematical Finance ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Taylor Approximation ◽  
Wide Range ◽  
Delay Differential

Stochastic delay differential equations with jumps have a wide range of applications, particularly, in mathematical finance. Solution of the underlying initial value problems is important for the understanding and control of many phenomena and systems in the real world. In this paper, we construct a robust Taylor approximation scheme and then examine the convergence of the method in a weak sense. A convergence theorem for the scheme is established and proved. Our analysis and numerical examples show that the proposed scheme of high order is effective and efficient for Monte Carlo simulations for jump-diffusion stochastic delay differential equations.

scholarly journals Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanqiang Chang ◽  
Huabin Chen
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Markovian Switching ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Almost Surely Exponential Stability ◽  
Delay Differential

<p style='text-indent:20px;'>In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th(<inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>) for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M4">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.</p>

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