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Theory of Probability and Mathematical Statistics

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Difference approximation of the local times of multidimensional diffusions

Author: Aleksey M. Kulik
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 97-114
MSC (2000): Primary 60J55, 60J45, 60F17
Published electronically: August 4, 2009
MathSciNet review: 2446852
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Abstract: Sequences of additive functionals of difference approximations are considered for multidimensional uniformly nondegenerate diffusions. Sufficient conditions are obtained for the weak convergence of such sequences to a $ W$-functional of the limit process. The class of $ W$-functionals appearing as limits for such a problem can be described uniquely in terms of the corresponding $ W$-measures $ \mu$ as follows:

$\displaystyle \lim_{\delta\downarrow 0}\sup_{x\in\mathbb{R}^m}\int_{\Vert y-x\Vert\leq \delta}w(\Vert y-x\Vert) \mu(dy)=0, $

where $ w(r)=\begin{cases}\max(-\ln r, 1),& m=2,\\ r^{2-m},& m>2. \end{cases}$

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Additional Information

Aleksey M. Kulik
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street 3, 01601, Kyiv, Ukraine

Keywords: Additive functional, local time, characteristic, $W$-measure, Markov approximation
Received by editor(s): February 12, 2007
Published electronically: August 4, 2009
Additional Notes: Supported by the Ministry of Education and Science of Ukraine, project GP/F13/0095
Article copyright: © Copyright 2009 American Mathematical Society