On the almost sure running maxima of solutions of affine stochastic functional differential equations

John A.D. Appleby, Xuerong Mao, H. Wu

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This paper studies the large fluctuations of solutions of scalar and finite-dimensional affine stochastic functional differential equations with finite memory as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process.
Original languageEnglish
Pages (from-to)646-678
Number of pages33
JournalSIAM Journal on Mathematical Analysis
Issue number2
Early online date31 Mar 2010
Publication statusPublished - 2010


  • stochastic functional differential equation
  • differential resolvent
  • stationary process
  • gaussian process
  • finite delay
  • asymptotic estimation
  • running maxima

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