Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, 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.