scholarly journals Almost Sure andLpConvergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Qian Guo ◽  
Xueyin Tao
Keyword(s):  
Differential Equation ◽  
Almost Sure Convergence ◽  
Euler Method ◽  
Backward Euler Method ◽  
Variable Delay ◽  
Cantelli Lemma ◽  
Stochastic Delay ◽  
Backward Euler ◽  
Stochastic Delay Differential Equation ◽  
Delay Differential

The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven inLp-sense. Almost sure convergence is derived from theLpconvergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.

Split-Step Backward Euler Method for Stochastic Delay Hopfield Neural Networks with Markovian Switching

2011 ◽  
Vol 58-60 ◽  
pp. 1390-1395
Author(s):  
Rong Hua Li ◽  
Li Yang ◽  
Jia Wei Li
Keyword(s):  
Neural Networks ◽  
Euler Method ◽  
Hopfield Neural Networks ◽  
Markovian Switching ◽  
Backward Euler Method ◽  
Approximation Solution ◽  
True Solution ◽  
Stochastic Delay ◽  
Backward Euler ◽  
Exponentially Stable

In this paper, split-step backward Euler method for stochastic delay Hopfield neural networks with Markovian switching is considered. The main aim of this paper is to show that the numerical approximation solution is convergent to the true solution with order. The conditions under which the numerical solution is exponentially stable in mean square are given. An example is provided for illustration.

scholarly journals The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qian Guo ◽  
Xueyin Tao
Keyword(s):  
Differential Equation ◽  
Numerical Experiments ◽  
Time Varying ◽  
Almost Sure Stability ◽  
Stochastic Delay ◽  
Almost Sure Exponential Stability ◽  
Time Varying Delay ◽  
Backward Euler ◽  
Delay Differential ◽  
Varying Delay

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

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