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Large deviation theory for stochastic difference equations

Published online by Cambridge University Press:  01 December 1997

R. KUSKE  and
J. B. KELLER
R. KUSKE
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455, USA
Supported in part by a NSF Mathematical Sciences Postdoctoral Fellowship.
J. B. KELLER
Affiliation:
Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
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Abstract

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The probability density for the solution yn of a stochastic difference equation is considered. Following Knessl et al. [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 8 , Issue 6 , December 1997 , pp. 567 - 580
Copyright
1997 Cambridge University Press