79 research outputs found
Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation
This paper studies the asymptotic growth and decay properties of solutions of
the stochastic pantograph equation with multiplicative noise. We give
sufficient conditions on the parameters for solutions to grow at a polynomial
rate in -th mean and in the almost sure sense. Under stronger conditions the
solutions decay to zero with a polynomial rate in -th mean and in the almost
sure sense. When polynomial bounds cannot be achieved, we show for a different
set of parameters that exponential growth bounds of solutions in -th mean
and an almost sure sense can be obtained. Analogous results are established for
pantograph equations with several delays, and for general finite dimensional
equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of
Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ.
(July 1--4, 2015, Szeged, Hungary
Stability of numerical method for semi-linear stochastic pantograph differential equations
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
Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps
The sufficient conditions of existence and uniqueness of the solutions for nonlinear stochastic pantograph equations with Markovian switching and jumps are given. It is proved that Euler-Maruyama scheme for nonlinear stochastic pantograph equations with Markovian switching and Brownian motion is of convergence with strong order 1/2. For nonlinear stochastic pantograph equations with Markovian switching and pure jumps, it is best to use the mean-square convergence, and the order of mean-square convergence is close to 1/2
The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps
In this paper, we consider a class of stochastic pantograph differential equations with Lévy jumps (SPDEwLJs). By using the Burkholder-Davis-Gundy inequality and the Kunita's inequality, we prove the existence and uniqueness of solutions to SPDEwLJs whose coefficients satisfying the Lipschitz conditions and the local Lipschitz conditions. Meantime, we establish the p-th exponential estimations and almost surely asymptotic estimations of solutions to SPDEwLJs
Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations
The exponential stability of numerical methods to stochastic differential
equations (SDEs) has been widely studied. In contrast, there are relatively few
works on polynomial stability of numerical methods. In this letter, we address
the question of reproducing the polynomial decay of a class of SDEs using the
Euler--Maruyama method and the backward Euler--Maruyama method. The key
technical contribution is based on various estimates involving the gamma
function
The asymptotic stability of hybrid stochastic systems with pantograph delay and non-Gaussian Lévy noise
The main aim of this paper is to investigate the asymptotic stability of hybrid stochastic systems with pantograph delay and non-Gaussian Lévy noise (HSSwPDLNs). Under the local Lipschitz condition and non-linear growth condition, we investigate the existence and uniqueness of the solution to HSSwPDLNs. By using the Lyapunov functions and M-matrix theory, we establish some sufficient conditions on the asymptotic stability and polynomial stability for HSSwPDLNs. Finally, two examples are provided to illustrate our results
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