An estimate of the rate of convergence of a sequence of additive functionals of difference approximations for a multidimensional diffusion process

Author:
Iu. V. Ganychenko

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **90** (2014).

Journal:
Theor. Probability and Math. Statist. **90** (2015), 23-41

MSC (2010):
Primary 60J55; Secondary 60F17

DOI:
https://doi.org/10.1090/tpms/947

Published electronically:
August 6, 2015

MathSciNet review:
3241858

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a sequence of additive functionals of difference approximations for a multidimensional diffusion. A result by A. M. Kulik, *Difference approximation for local times of multidimensional diffusions*, Theory Probab. Math. Statist. **78** (2008), 67-83, on sufficient conditions for such a sequence to converge weakly to a -functional of the limit process is improved. An estimate of the rate of convergence is obtained.

**1.**I. I. Gikhman and A. V. Skorokhod,*Introduction to the Theory of Random Processes*, ``Nauka'', Moscow, 1977; English transl., W. B. Saunders, Philadelphia, 1969. MR**0247660 (40:923)****2.**D. Gusak, A. Kukush, A. Kulik, Y. Mishura, and A. Pilipenko,*Theory of Stochastic Processes With Applications to Financial Mathematics and Risk Theory*, Kyiv University Press, Kyiv, 2008; English transl., Springer, Berlin, 2010. MR**2572942 (2011f:60069)****3.**E. B. Dynkin,*Markov Processes*, ``Fizmatgiz'', Moscow, 1963; English transl., Academic Press, Inc., New York, 1965. MR**0193670 (33:1886)****4.**Yu. N. Kartashov and A. M. Kulik,*Weak convergence of additive functionals of a sequence of Markov chains*, Theory Stoch. Process.**15 (31)**(2009), no. 1, 15-32. MR**2603167 (2011a:60130)****5.**V. Konakov and E. Mammen,*Local limit theorems for transition densities of Markov chains converging to diffusions*, Probab. Theory Rel. Fields**117**(2000), 551-587. MR**1777133 (2001j:60141)****6.**V. Konakov,*Small time asymptotics in local limit theorems for Markov chains converging to diffusions*, arxiv:math. PR/0602429, 2006.**7.**A. M. Kulik,*Additive functionals of Markov processes and local times of stochastic processes*, Matematika segodnya (2009), 39-66. (Russian)**8.**A. M. Kulik,*Difference approximation for local times of multidimensional diffusions*, Theory Probab. Math. Statist.**78**(2008), 67-83. MR**2446852 (2010b:60212)****9.**A. M. Kulik,*Markov approximation of stable processes by random walks*, Theory Stoch. Process.**12 (28)**(2006), no. 1-2, 87-93. MR**2316289 (2008j:60082)****10.**A. V. Skorokhod,*Asymptotic Methods in the Theory of Stochastic Differential Equations*, ``Naukova Dumka'', Kiev, 1987; English transl., American Mathematical Society, Providence, 2008. MR**913305 (88m:60164)****11.**A. V. Skorokhod,*Lectures on the Theory of Stochastic Processes*, ``Lybid''', Kyiv, 1990; English transl., VSP/TViMS, Utrecht/Kyiv, 1996. MR**1452108 (99d:60001)**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60J55,
60F17

Retrieve articles in all journals with MSC (2010): 60J55, 60F17

Additional Information

**Iu. V. Ganychenko**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine

Email:
iurii_ganychenko@ukr.net

DOI:
https://doi.org/10.1090/tpms/947

Keywords:
Additive functionals,
characteristic of an additive functional,
$W$-measure,
Markov approximation,
diffusion process,
local time,
rate of convergence

Received by editor(s):
April 25, 2013

Published electronically:
August 6, 2015

Article copyright:
© Copyright 2015
American Mathematical Society