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A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

  • Scott Hottovy
  • Austin McDanielEmail author
  • Jan Wehr
Article
  • 14 Downloads

Abstract

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein–Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

Keywords

Stochastic differential delay equations Noise-induced drift Itô–Stratonovich transition 

Notes

Acknowledgements

A.M. and J.W. were partially supported by the NSF Grants DMS 1009508 and DMS 0623941.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUnited States Naval AcademyAnnapolisUSA
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA
  3. 3.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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