Stability of Stochastic Functional Differential Equations with Regime-Switching: Analysis Using Dupire's Functional Ito Formula

This work focuses on almost sure andL(p)stability of stochastic functional differential equations by using Lyapunov functionals with the help of the recently developed Dupire's functional Ito formula. Novel conditions for stability, which are different from those in the existing literature, are given in terms of Lyapunov functionals. It is demonstrated that the conditions are useful for stochastic stabilization. It is also shown that adding a diffusion term can stabilize an unstable system of deterministic differential equations with Markov switching. Furthermore, a robustness result is obtained, which states that the stability of stochastic differential equations with regime-switching is preserved under delayed perturbations when the delay is small enough.