Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Asymptotic expansions of solutions of systems of Kolmogorov backward equations for two-time-scale switching diffusions


Authors: Dung Tien Nguyen and G. Yin
Journal: Quart. Appl. Math. 71 (2013), 601-628
MSC (2010): Primary 34E05, 60J27, 93E20
DOI: https://doi.org/10.1090/S0033-569X-2013-01277-X
Published electronically: August 27, 2013
MathSciNet review: 3136987
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Abstract: This work is concerned with systems of coupled partial differential equations (known as Kolmogorov backward equations) for continuous-time Markov processes featuring in the coexistence of continuous dynamics and discrete events. Arising from state-dependent switching diffusions, distinct from the usual Markovian regime-switching systems, the generator of the switching component depends on the continuous state. One of the main ingredients of our models is the two-time-scale formulation. In contrast to the work on Kolmogorov forward equations in the existing literature, new techniques are developed in this paper. Although they originate from probabilistic models, the methods are analytic. Two classes of models, namely, fast-switching systems and fast-diffusion systems, are treated. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations. These asymptotic series are rigorously justified and error bounds are obtained.


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Additional Information

Dung Tien Nguyen
Affiliation: Department of Applied Mathematics, Faculty of Applied Science, University of Technology, Ho Chi Minh city, Vietnam
Email: dungnt@hcmut.edu.vn

G. Yin
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: gyin@math.wayne.edu

DOI: https://doi.org/10.1090/S0033-569X-2013-01277-X
Keywords: System of backward equation, two-time scale, asymptotic expansion.
Received by editor(s): May 15, 2011
Published electronically: August 27, 2013
Additional Notes: The research of the first author was supported in part by Wayne State University under a Graduate Research Assistantship.
The research of the second author was supported in part by the National Science Foundation under DMS-0603287 and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210.
Article copyright: © Copyright 2013 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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