ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150010 ◽  
Author(s):  
N. SRI NAMACHCHIVAYA ◽  
VOLKER WIHSTUTZ
Keyword(s):  
Asymptotic Stability ◽  
Markov Process ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Almost Sure Stability ◽  
Stochastic Delay ◽  
Finite Dimensional ◽  
Finite State ◽  
Parametric Fluctuations ◽  
Delay Differential

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

scholarly journals Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching

2021 ◽  
Vol 19 (1) ◽  
pp. 614-628
Author(s):  
Xiaozhi Zhang ◽  
Zhangsheng Zhu ◽  
Chenggui Yuan
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Stability Of Solutions ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
The Stability ◽  
Delay Differential

Abstract The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.

scholarly journals Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Zhanhua Yu ◽  
Mingzhu Liu
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Backward Euler ◽  
Delay Differential ◽  
Theoretical Results

We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our 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.

Sign in / Sign up

Export Citation Format

Share Document