# American Institute of Mathematical Sciences

June  2008, 21(2): 571-593. doi: 10.3934/dcds.2008.21.571

## Almost sure stability of some stochastic dynamical systems with memory

 1 Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7 2 Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316 3 Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str., 163-a, Donetsk, 83015, Ukraine

Received  April 2007 Revised  January 2008 Published  March 2008

Almost sure asymptotic stability of stochastic difference and differential equations with non-anticipating memory terms is studied in $\R^1$. Sufficient criteria are obtained with help of Lyapunov-Krasovskiĭ-type functionals, martingale decomposition and semi-martingale convergence theorems. The results allow numerical methods for stochastic differential equations with memory to be studied in terms of their ability to reproduce almost sure stability.
Citation: Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571
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