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

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.