ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS

Abstract

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.