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

Results are presented on the stability of solutions of stochastic delay differential equations with multiplicative noise and of convergent numerical solutions obtained by a a method of Euler-Maruyama type. An attempt is made to provide a fairly self-contained presentation. A basic concept of the stability of a solution of an evolutionary stochastic delay differential equation is concerned with the sensitivity of the solution to perturbations in the initial function. We recall the stability definitions considered herein and show that an inequality of Halanay type (derivable via comparison theory)j and deterministic results can be employed to derive stability conditions for solutions of suitable equations. In practice, dosed form solutions of stochastic delay differential equations are unlikely to he available. In the second part of the paper a stability theory for numerical solutions (solutions of Euler type) is considered. A convergence result is recalled for completeness and new stability results are obtained using a discrete analogue of the continuous Halanay-type inequality and results for a deterministic recurrence relation. Various results for stochastic (ordinary) differential equations with no time lag or for deterministic delay differential equations can he deduced from the results given here.