Difference approximation of the local times of multidimensional diffusions

Kulik, Aleksey

doi:10.1090/S0094-9000-09-00765-0

Difference approximation of the local times of multidimensional diffusions

Author: Aleksey M. Kulik
Translated by: S. Kvasko
Journal: Theor. Probability and Math. Statist. 78 (2009), 97-114
MSC (2000): Primary 60J55, 60J45, 60F17
DOI: https://doi.org/10.1090/S0094-9000-09-00765-0
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: \[ \lim _{\delta \downarrow 0}\sup _{x\in \mathbb {R}^m}\int _{\|y-x\|\leq \delta }w(\|y-x\|) \mu (dy)=0, \] where $w(r)=\begin {cases} \max (-\ln r, 1),& m=2,\\ r^{2-m},& m>2. \end {cases}$

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