Evelyn Buckwar, Christopher T.H. Baker,
"Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations"
, in Journal of Computational and Applied Mathematics, Vol. 184, Nummer 2, Elsevier Science B.V. (North-Holland), Amsterdam, Seite(n) 404-427, 2005, ISSN: 0377-0427
Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations
Sprache des Titels:
One concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data; there are other stability concepts, notably those concerned with persistent perturbations. Results are presented on the stability in p-th mean of solutions of stochastic delay differential equations with multiplicative noise, and of stochastic delay difference equations. The difference equations are of a type found in numerical analysis and we employ our results to obtain mean-square stability criteria for the solution of the Euler?Maruyama discretization of stochastic delay differential equations.
The analysis proceeds as follows: We show that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution. We then produce a discrete analogue of the Halanay-type theory, that permits us to develop a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.