Stochastic control based on time-change transformations for stochastic processes with Lévy noise
Authors:
S. V. Bodnarchuk and O. M. Kulik
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 86 (2013), 13-31
MSC (2010):
Primary 60H07; Secondary 60G51
DOI:
https://doi.org/10.1090/S0094-9000-2013-00886-2
Published electronically:
August 20, 2013
MathSciNet review:
2986447
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We propose a new method of stochastic control for stochastic processes with Lévy noise based on time-change transformations. Applying this method, we prove that the integral minorization condition holds for Markov processes defined by stochastic equations with Lévy noise and obtain the explicit estimates for the rate of convergence in the ergodic theorem.
References
- Alexey M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stochastic Process. Appl. 119 (2009), no. 2, 602–632. MR 2494006, DOI https://doi.org/10.1016/j.spa.2008.02.006
- Alexey M. Kulik, Absolute continuity and convergence in variation for distributions of functionals of Poisson point measure, J. Theoret. Probab. 24 (2011), no. 1, 1–38. MR 2782709, DOI https://doi.org/10.1007/s10959-010-0325-4
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- R. L. Dobrušin, Definition of a system of random variables by means of conditional distributions, Teor. Verojatnost. i Primenen. 15 (1970), 469–497 (Russian, with English summary). MR 0298716
- A. Yu. Veretennikov, Coupling method for Markov chains under integral Doeblin type condition, Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), 2002, pp. 383–390. MR 2027410
- L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
- Peter Lancaster, Theory of matrices, Academic Press, New York-London, 1969. MR 0245579
References
- A. M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stoch. Process. Appl. 119 (2009), no. 2, 602–632. MR 2494006 (2010i:60176)
- A. M. Kulik, Absolute continuity and convergence in variation for distributions of functionals of Poisson point measure, J. Theor. Probab. 24 (2011), no. 1, 1–38. MR 2782709 (2012e:60154)
- G. Federer, Geometric Measure Theory, Springer, New York, 1969. MR 0257325 (41:1976)
- R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Teor. Veroyatnost. Primenen. 15 (1970), no. 3, 469–497; English transl. in Theory Probab. Appl. 15 (1970), no. 3, 458–486. MR 0298716 (45:7765)
- A. Yu. Veretennikov, Coupling method for Markov chains under integral Doeblin type condition, Theory Stoch. Process. 8(24) (2002), no. 3–4, 383–391. MR 2027410 (2004j:60151)
- L. Nierenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. MR 0488102 (58:7672)
- P. Lankaster, Theory of Matrices, Academic Press, New York–London, 1969. MR 0245579 (39:6885)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H07,
60G51
Retrieve articles in all journals
with MSC (2010):
60H07,
60G51
Additional Information
S. V. Bodnarchuk
Affiliation:
National Technical University of Ukraine “KPI”, Peremogy Avenue 37, 03056, Kyiv, Ukraine
Email:
sem_bodn@ukr.net
O. M. Kulik
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs′ka Street 3, 01601, Kyiv, Ukraine
Email:
kulik@imath.kiev.ua
Keywords:
Lévy process,
stochastic equation,
stochastic control,
time-change transformations
Received by editor(s):
December 5, 2011
Published electronically:
August 20, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
