Almost sure exponential stability of stochastic differential delay equations

Qian Guo, Xuerong Mao, Rongxian Yue

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)
185 Downloads (Pure)


This paper is concerned with the almost sure exponential stability of the multidimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form dx(t) = f(x(t−δ1(t)), t)dt+g(x(t−δ2(t)), t)dB(t), where δ1, δ2 : R+ → [0, τ ] stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) dy(t) = f(y(t), t)dt + g(y(t), t)dB(t)
admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number τ ∗
such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by τ ∗ . We provide an implicit lower bound for τ ∗ which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equations
Original languageEnglish
Pages (from-to)1919-1933
Number of pages15
JournalSIAM Journal on Control and Optimization
Issue number4
Publication statusPublished - 27 Jul 2016


  • almost sure exponential stability
  • stochastic differential delay equations
  • Ito formula
  • Brownian motion
  • stochastic stabilization

Cite this