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Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes
Author(s) -
Danilov Vladimir G.,
Frolovitchev Serguei M.
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200007)215:1<55::aid-mana55>3.0.co;2-r
Subject(s) - mathematics , markov chain , statistical physics , markov process , statistics , physics
In this paper we consider some Kolmogorov–Feller equations with a small parameter h . We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h . This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h . We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.