Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations

• Published in: Numerical algorithms Vol. 80; no. 4; pp. 1391 - 1411
• Main Authors: Guo, Ping, Li, Chong–Jun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2019; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Decay rate; Differential equations; Exact solutions; Liapunov exponents; Numeric Computing; Numerical Analysis; Numerical methods; Original Paper; Pantographs; Polynomials; Stability; Theory of Computation; Stochastic linear theta method; 60H10; General decay rate; 65C30; 65L20; Stochastic pantograph differential equation; Almost sure stability; Split-step theta method
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-018-0531-1

The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for 𝜃 ∈ 0 , 1 2 are stronger than those for 𝜃 ∈ 1 2 , 1 . The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability.