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 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 Some Properties of Numerical Solutions for Semilinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

2021 ◽  
Vol 2021 ◽  
pp. 1-26
Author(s):  
Haiyan Yuan
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Exponential Euler Method ◽  
The Stability ◽  
Delay Differential

This paper is concerned with the numerical solutions of semilinear stochastic delay differential equations driven by G-Brownian motion (G-SLSDDEs). The existence and uniqueness of exact solutions of G-SLSDDEs are studied by using some inequalities and the Picard iteration scheme first. Then the numerical approximation of exponential Euler method for G-SLSDDEs is constructed, and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent, and it can reproduce the stability of the analytical solution under some restrictions. Numerical experiments are presented to confirm the theoretical 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.

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