Semi-Markov approach to the problem of delayed reflection of diffusion Markov processes
Author:
B. P. Harlamov
Journal:
Theor. Probability and Math. Statist. 89 (2014), 13-22
MSC (2010):
Primary 60J25, 60J60
DOI:
https://doi.org/10.1090/S0094-9000-2015-00931-5
Published electronically:
January 26, 2015
MathSciNet review:
3235171
Full-text PDF Free Access
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Abstract: A one-dimensional diffusion process taking positive values and reflecting from zero is considered. All the variants of reflecting that preserve of the semi-Markov property are described. This property is characterized by a family of Laplace images of times between the first hitting of zero and first hitting of a level $r$ for any $r>0$. The parameter of this family is used to construct a time change transforming a process with instantaneous reflection to the process with delayed reflection.
References
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- Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated from the German by John R. Aumann et al. MR 0141601
- B. P. Kharlamov, A diffusion process with delay at the ends of a segment, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 351 (2007), no. Veroyatnost′i Statistika. 12, 284–297, 303–304 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 152 (2008), no. 6, 958–965. MR 2742915, DOI https://doi.org/10.1007/s10958-008-9114-3
- Boris Harlamov, Continuous semi-Markov processes, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2376500
- B. P. Kharlamov, On a Markov diffusion process with delayed reflection from an interval boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 368 (2009), no. Veroyatnost′i Statistika. 15, 243–267, 287 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 167 (2010), no. 4, 574–587. MR 2749196, DOI https://doi.org/10.1007/s10958-010-9945-6
- S. S. Rasova and B. P. Kharlamov, On the motion of Brownian particles along a delaying screen, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 396 (2011), no. Veroyatnost′i Statistika. 17, 175–194, 260–261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 188 (2013), no. 6, 737–747. MR 2870140, DOI https://doi.org/10.1007/s10958-013-1165-4
- B. P. Harlamov, Stochastic model of gas capillary chromatography, Comm. Statist. Simulation Comput. 41 (2012), no. 7, 1023–1031. MR 2912932, DOI https://doi.org/10.1080/03610918.2012.625782
References
- I. I. Gihman and A. V. Skorokhod, Stochastic Differential Equation, “Naukova Dumka”, Kiev, 1968; English transl. Springer-Verlag, Berlin, 1972. MR 0346904 (49 \#11625)
- F. Hausdorff, Set Theory, Chelsea Publishing Co., New York, 1962. (translated from the German) MR 0141601 (25:4999)
- B. P. Harlamov, Diffusion processes with delay at the endpoints of a segment, Zapiski Nauchnyh Seminarov POMI 351 (2007), 284–297; English transl. in J. Math. Sci. 152 (2008), no. 6, 958–965. MR 2742915 (2011j:60267)
- B. P. Harlamov, Continuous Semi-Markov Processes, Applied Stochastic Methods Series. ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2376500 (2009b:60275)
- B. P. Harlamov, On Markov diffusion processes with delayed reflection from boundaries of a segment, Zapiski Nauchnyh Seminarov POMI 368 (2009), 243–267; English transl. in J. Math. Sci. 167, (2010), no. 4, 574–587. MR 2749196 (2012b:60261)
- S. S. Rasova and B. P. Harlamov, On motion of Brownian particles along a delaying screen, Zapiski Nauchnyh Seminarov POMI 396 (2011), 175–194; English transl. in J. Math. Sci. 188 (2013), no. 6, 737–747. MR 2870140
- B. P. Harlamov, Stochastic model of gas capillary chromatography, Comm. Statist. Simulation Comput. 41 (2012), no. 7, 1023–1031. MR 2912932
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Additional Information
B. P. Harlamov
Affiliation:
Institute of Problems of Mechanical Engineering, RAS, Saint-Petersburg, Russia
Email:
b.p.harlamov@gmail.com
Keywords:
Diffusion,
Markov property,
continuous semi-Markov processes,
reflection,
delaying,
first exit time,
transition function,
Laplace transform,
change of time
Received by editor(s):
January 30, 2013
Published electronically:
January 26, 2015
Additional Notes:
This work was supported by grant RFBR 12-01-00457-a
Article copyright:
© Copyright 2015
American Mathematical Society
