Almost sure exponential stability of hybrid stochastic functional differential equations

This paper is concerned with the almost sure exponential stability of the n-dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ₁(x_t,t),r(t),t)dt+g(ψ₂(x_t,t),r(t),t)dB(t), where x_t={x(t+u):−τ≤u≤0} is a C([−τ,0];Rⁿ)-valued process, B(t) is an m-dimensional Brownian motion while r(t) is a Markov chain. We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t)=f(y(t),r(t),t)dt+g(y(t),r(t),t)dB(t) is almost surely exponentially stable, then there exists a positive number τ^⁎ such that the SFDE is also almost surely exponentially stable as long as τ<τ^⁎. We also describe a method to determine τ^⁎ which can be computed numerically in practice.