Abstract
As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size $h>0$ . Numerical examples further illustrate the obtained theoretical results.
Highlights
Stochastic delay differential equations played an important role in application areas, such as physics, biology, economics, and finance [ – ]
This paper mainly proves that if an analytical solution is stable, so is the exponential Euler method applied to the system for any step-size h >
We introduce the exponential Euler method for semi-linear stochastic pantograph differential equations
Summary
Stochastic delay differential equations played an important role in application areas, such as physics, biology, economics, and finance [ – ]. Fan [ ] investigated mean-square asymptotic stability of the θ method for linear stochastic pantograph differential equations. Xiao [ ] proved mean-square stability of the Milstein method for stochastic pantograph differential equations under suitable conditions. The exponential Euler method for semi-linear stochastic pantograph differential equations is general mean-square stable. We introduce the exponential Euler method for semi-linear stochastic pantograph differential equations . For any step-size h > , if the exponential Euler method to equation ( ) generates a numerical approximation that satisfies lim n→∞. Suppose that the conditions ( ) and ( ) hold, for arbitrary h > , the numerical solution of the exponential Euler method is general mean-square stable. The numerical solution produced by the exponential Euler method is mean-square stable for any step-size h >
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