Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Authors:
Fuke Wu;Xuerong Mao;Lukas Szpruch
Affiliations:
Huazhong University of Science and Technology, School of Mathematics and Statistics, 430074, Wuhan, Hubei, People’s Republic of China;University of Strathclyde, Department of Mathematics and Statistics, G1 1XH, Glasgow, UK;University of Strathclyde, Department of Mathematics and Statistics, G1 1XH, Glasgow, UK
Venue:
Numerische Mathematik
Year:
2010

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Abstract

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.