A note on exponential stability in th mean of solutions of stochastic delay differential equations

In a very recent paper, Baker and Buckwar [Exponential stability in p th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math. 184 (2005) 404–427] investigated the exponential stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise, and of stochastic difference equations which are Euler–Maruyama discretization of stochastic delay differential equations. The Dini derivative of the expectation of V ( t , X ( t ) ) “along” X ( t ) is taken in their stability analysis. Unfortunately, the main results derived by them are somewhat restrictive to be applied for practical purposes. In this note we shall improve the corresponding results. An example is given to illustrate our theory.