Theory of Probability and Mathematical Statistics
An estimate of the rate of convergence of a sequence of additive functionals of difference approximations for a multidimensional diffusion process
Iu. V. Ganychenko
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.
Keywords: Additive functionals, characteristic of an additive functional, W-measure, Markov approximation, diffusion process, local time, rate of convergence
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.
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.
3. E. B. Dynkin, Markov Processes, Fizmatgiz'', Moscow, 1963; English transl., Academic Press, Inc., New York, 1965.
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.
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.
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.
9. A. M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12 (28) (2006), no. 1-2, 87-93.
10. A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Naukova Dumka'', Kiev, 1987; English transl., American Mathematical Society, Providence, 2008.
11. A. V. Skorokhod, Lectures on the Theory of Stochastic Processes, ''Lybid'', Kyiv, 1990; English transl., VSP/TViMS, Utrecht/Kyiv, 1996.