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
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 Free Access
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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 $W$-functional of the limit process is improved. An estimate of the rate of convergence is obtained.
References
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- Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, and Andrey Pilipenko, Theory of stochastic processes, Problem Books in Mathematics, Springer, New York, 2010. With applications to financial mathematics and risk theory. MR 2572942
- E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
- Yuri N. Kartashov and Alexey M. Kulik, Weak convergence of additive functionals of a sequence of Markov chains, Theory Stoch. Process. 15 (2009), no. 1, 15–32. MR 2603167
- Valentin Konakov and Enno Mammen, Local limit theorems for transition densities of Markov chains converging to diffusions, Probab. Theory Related Fields 117 (2000), no. 4, 551–587. MR 1777133, DOI https://doi.org/10.1007/PL00008735
- V. Konakov, Small time asymptotics in local limit theorems for Markov chains converging to diffusions, arxiv:math. PR/0602429, 2006.
- A. M. Kulik, Additive functionals of Markov processes and local times of stochastic processes, Matematika segodnya (2009), 39–66. (Russian)
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References
- 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)
- 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)
- E. B. Dynkin, Markov Processes, “Fizmatgiz”, Moscow, 1963; English transl., Academic Press, Inc., New York, 1965. MR 0193670 (33:1886)
- 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)
- 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)
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- A. M. Kulik, Additive functionals of Markov processes and local times of stochastic processes, Matematika segodnya (2009), 39–66. (Russian)
- A. M. Kulik, Difference approximation for local times of multidimensional diffusions, Theory Probab. Math. Statist. 78 (2008), 67–83. MR 2446852 (2010b:60212)
- 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)
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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
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
