Open AccessAlmost sure exponential stability of stochastic differential delay equations
Author(s) -
Wei Zhang,
Minghui Song,
M.Z. Liu
Publication year - 2019
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1903789z
Subject(s) - mathematics , exponential stability , lipschitz continuity , moment (physics) , stochastic differential equation , exponential function , exponential growth , mathematical analysis , stability (learning theory) , lyapunov function , second moment of area , geometry , nonlinear system , physics , computer science , classical mechanics , quantum mechanics , machine learning
This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same results. Hence, our new theory enables us to consider the almost sure exponential stability of the SDDEs using the stochastic theta method, instead of the method of Lyapunov functions. That is, we can now perform careful numerical simulations using the stochastic theta method with a sufficiently small step size ?t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ? (0,1), we can then deduce that the underlying SDDE is almost sure exponentially stable. Our new theory also enables us to show the pth moment exponential stability of the stochastic theta method to reproduce the almost sure exponential stability of the SDDEs.