Existence of a limit distribution of a solution of a linear inhomogeneous stochastic differential equation
Author:
D. O. Ivanenko
Translated by:
N. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal:
Theor. Probability and Math. Statist. 78 (2009), 4960
MSC (2000):
Primary 60F05; Secondary 60J75
Published electronically:
August 4, 2009
MathSciNet review:
2446848
Fulltext PDF Free Access
Abstract 
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Abstract: We find conditions for the existence of a limit distribution (as ) of a vector process defined in and determined by an inhomogeneous stochastic differential equation , where is a nonrandom continuous increasing function, and are independent Poisson and centered Poisson measures, respectively.
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 I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications, Naukova dumka, Kiev, 1982. (Russian) MR 678374 (84j:60003)
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 O. K. Zakusilo, On classes of limit distributions in a summation scheme, Teor. Verojatnost. i Mat. Statist. 12 (1975), 6269; English transl. in Theory Probab. Math. Statist. 12 (1976), 4448. MR 0397833 (53:1689)
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 O. K. Zakusilo, Some properties of random vectors of the form , Teor. Verojatnost. i Mat. Statist. 13 (1975), 5962; English transl. in Theory Probab. Math. Statist. 13 (1976), 6264. MR 0415734 (54:3814)
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 V. V. Anisimov, O. K. Zakusilo, and V. S. Donchenko, Elements of Queueing Theory and Asymptotic Analysis of Systems, Vyshcha shkola, Kiev, 1987. (Russian)
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 H. Masuda, On multidimensional OrnsteinUhlenbeck processes driven by a general Lévy process, Bernoulli 10 (2004), no. 1, 97120. MR 2044595 (2004m:60080)
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 H. Masuda, Ergodicity and exponentialmixing bounds for multidimensional diffusions with jumps, Stoch. Proc. Appl. 117 (2007), 3556. MR 2287102 (2008j:60194)
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 K. Sato and M. Yamazato, Operator selfdecomposable distributions as limit distributions of processes of OrnsteinUhlenbeck type, Stoch. Proc. Appl. 17 (1984), 73100. MR 738769 (86j:60048)
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Additional Information
D. O. Ivanenko
Affiliation:
Department of Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
ida@univ.kiev.ua
DOI:
http://dx.doi.org/10.1090/S0094900009007613
PII:
S 00949000(09)007613
Keywords:
Limit distribution,
Poisson measure,
It\^o's formula,
Tauberian theorem
Received by editor(s):
July 3, 2007
Published electronically:
August 4, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
