Diffusion processes in a composite environment
Author:
S. Ya. Makhno
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 94 (2017), 137-149
MSC (2010):
Primary 60J60, 60J55, 60J65
DOI:
https://doi.org/10.1090/tpms/1014
Published electronically:
August 25, 2017
MathSciNet review:
3553459
Full-text PDF
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Additional Information
Abstract: Sufficient conditions for the weak convergence of solutions of stochastic equations in a space with barriers are obtained. It is assumed that the number of barriers grows to infinity and in the limit they fill an entire segment. An equation for the limit process is given.
References
- Kôsaku Yosida, Functional analysis, 4th ed., Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 123. MR 0350358
- J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, Stochastic analysis and applications (Swansea, 1983) Lecture Notes in Math., vol. 1095, Springer, Berlin, 1984, pp. 51–82. MR 777514, DOI https://doi.org/10.1007/BFb0099122
- S. Ya. Makhno, Stochastic Equations. Limit Theorems, “Naukova dumka”, Kyiv, 2012. (Russian)
- M. I. Portenko, Protsesi difuzīï v seredovishchakh z membranami, Trudi Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni [Proceedings of the Institute of Mathematics of the National Academy of Sciences of the Ukraine], vol. 10, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 1995 (Ukrainian, with Ukrainian summary). MR 1356720
- S. B. Stechkin, A refinement of a proof in the book by V. I. Glivenko “Stieltjes integral”, Uspehi Matem. Nauk 3b (1948), no. 6, 213–215. (Russian)
References
- K. Iosida, Functional Analysis, Springer, Berlin–Heidelberg, 1965. MR 0350358
- J. F. Le Gall, On one dimensional stochastic equations involving local time of unknown processes, Stochastic Analysis and Applications (Swansea, 1983), Lecture Notes Math., vol. 1095, 51–82, Springer, Berlin, 1984. MR 777514
- S. Ya. Makhno, Stochastic Equations. Limit Theorems, “Naukova dumka”, Kyiv, 2012. (Russian)
- N. I. Portenko, Diffusion Processes in Environments with Membranes, Institute of Mathematics, National Academy of Science of Ukraine, Kiev, 1995. (Ukrainian) MR 1356720
- S. B. Stechkin, A refinement of a proof in the book by V. I. Glivenko “Stieltjes integral”, Uspehi Matem. Nauk 3b (1948), no. 6, 213–215. (Russian)
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Additional Information
S. Ya. Makhno
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 01030 Kyiv, Ukraine
Email:
smakhno@gmail.com
Keywords:
Stochastic equation,
local time of a process,
limit theorem,
weak convergence
Received by editor(s):
June 9, 2015
Published electronically:
August 25, 2017
Additional Notes:
The author was supported by grant # 09-01-14 from NANU–RFFI
Article copyright:
© Copyright 2017
American Mathematical Society
