Journal of the
Korean Mathematical Society JKMS

This work focuses on the general decay stability of nonlinear stochastic differential equations with unbounded delay. A Razumikhin-type theorem is first established to obtain the moment stability but without almost sure stability. Then an improved edition is presented to derive not only the moment stability but also the almost sure stability, while existing Razumikhin-type theorems aim at only the moment stability. By virtue of the $M$-matrix techniques, we further develop the aforementioned Razumikhin-type theorems to be easily implementable. Two examples are given for illustration.

Keywords: stochastic delay differential equation, unbounded delay, Razumikhin-type theorem, stability, $M$-matrix

MSC numbers: 60H10, 34K20, 34K50