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

Full-text PDF Free Access

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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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Additional Information

**Aleksey M. Kulik**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivsâ€™ka Street 3, 01601, Kyiv, Ukraine

Email:
kulik@imath.kiev.ua

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