Stability of stochastic functional differential systems using degenerate Lyapunov functionals and applications

Motivated by the seminal work of Dupire (2009) on functional Itô formulas, this work investigates asymptotic properties of systems represented by stochastic functional differential equations (SFDEs). Stability of general delay-dependent SFDEs is investigated using degenerate Lyapunov functionals, which are only positive semi-definite rather than positive definite as used in the classical work. This paper first establishes boundedness and regularity of SFDEs by using degenerate Lyapunov functionals. Then moment and almost sure exponential stabilities are obtained based on degenerate Lyapunov functionals and the semi-martingale convergence theorem. As an application of the stability criteria, consentability of stochastic multi-agent systems with nonlinear dynamics is studied.