Theory of Probability and Mathematical Statistics

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ISSN 1547-7363(e) ISSN 0094-9000(p)

Difference approximation of the local times of multidimensional diffusions

Author(s): Aleksey M. Kulik
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 78 (2008).
Journal: Theor. Probability and Math. Statist. No. 78 (2009), 97-114.
MSC (2000): Primary 60J55, 60J45, 60F17
Posted: August 4, 2009
MathSciNet review: 2446852
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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 -functional of the limit process. The class of -functionals appearing as limits for such a problem can be described uniquely in terms of the corresponding -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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