We consider a general stochastic differential delay equation (SDDE) with multiplicative colored noise. We study the limit as the time delays and the correlation times of the noises go to zero at the same rate. First, we derive the limiting equation for the equation obtained by Taylor expanding the SDDE to first order in the time delays. The limiting equation contains a noise-induced drift term that depends on the ratios of the time delays to the correlation times of the noises. We prove that, under appropriate assumptions, the solution of the equation obtained by the Taylor expansion converges to the solution of this limiting equation in probability with respect to the sup norm over compact time intervals. Next, we derive the limiting equation for the SDDE and prove a similar convergence result regarding convergence of the solution of the SDDE to the solution of this limiting equation. We see that the limiting equation corresponding to the equation obtained by the Taylor expansion is an approximation of the limiting equation corresponding to the SDDE. Finally, we study the effects of time delay on a particular model of active Brownian motion.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/556867 |
| Date | January 2015 |
| Creators | McDaniel, Austin James |
| Contributors | Wehr, Janek, Wehr, Janek, Kennedy, Tom, Sethuraman, Sunder, Ercolani, Nick |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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