Stability of the expected value and boundness of the variance of some stochastic time-delay differential equation and its application to the hemapoietic stem cell regulation system

The results of work of J. Lei and M. Mackey (SIAM J. Appl. Math., 67, 387-407 (2007)) will be presented. We will study the stability of the trivial solution of a stochastic delay differential equation in the presence of an additive and multiplicative white noise. We will show the conditions guaranteeing that the expected value of the solution converges to zero and that the variance is bounded. We will also give a condition for unboudness of the variance. The results will be applied to the hemapoietic stem cell regulation system.