Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

  • Authors:

  • Christopher T. H. Baker;Evelyn Buckwar

  • Affiliations:

  • Department of Mathematics, University College Chester, Chester, UK and School of Mathematics, The University of Manchester, Manchester, UK;Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden, Berlin, Germany

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2005

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Abstract

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.