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On Some Degenerate Large Deviation Problems
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Anatolii A. Puhalskii, University of Colorado at Denver, USA and Institute for Problems in Information,
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Abstract
This paper concerns the issue of obtaining the large deviation
principle for solutions of stochastic equations with possibly degenerate
coefficients. Specifically, we explore the potential of
the methodology that consists in
establishing exponential tightness and identifying the action
functional via a maxingale problem.
In the author's earlier work it has been demonstrated that certain
convergence properties
of the predictable characteristics of semimartingales
ensure both that exponential tightness holds and that every large deviation
accumulation point is a solution to a maxingale problem.
The focus here is on the uniqueness for the maxingale problem.
It is first shown that under certain continuity
hypotheses existence and uniqueness of a solution
to a maxingale problem of diffusion type are equivalent to
Luzin weak existence and uniqueness, respectively,
for the associated idempotent Ito equation.
Consequently, if the idempotent equation
has a unique Luzin weak solution, then the action functional is
specified uniquely, so the large deviation principle follows.
Two kinds of application are considered.
Firstly, we obtain results on the logarithmic asymptotics of
moderate deviations for stochastic
equations with possibly degenerate diffusion
coefficients which, as compared with
earlier results, relax the growth conditions on the
coefficients, permit certain non-Lipshitz-continuous
coefficients, and allow the coefficients
to depend on the entire past of the process and to be discontinuous
functions of time.
The other application concerns multiple-server queues with impatient customers.
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Full text: PDF
Pages: 862-886
Published on: December 31, 2004
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Electronic Journal of Probability. ISSN: 1083-6489 |
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