On the local time process of a skew Brownian motion
HTML articles powered by AMS MathViewer
- by Andrei Borodin and Paavo Salminen PDF
- Trans. Amer. Math. Soc. 372 (2019), 3597-3618 Request permission
Abstract:
We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.References
- Luis H. R. Alvarez E. and Paavo Salminen, Timing in the presence of directional predictability: optimal stopping of skew Brownian motion, Math. Methods Oper. Res. 86 (2017), no. 2, 377–400. MR 3717234, DOI 10.1007/s00186-017-0602-4
- Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, Edward Waymire, and Brian Wood, Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab. 21 (2011), no. 1, 183–214. MR 2759199, DOI 10.1214/10-AAP691
- Richard F. Bass and Philip S. Griffin, The most visited site of Brownian motion and simple random walk, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 417–436. MR 803682, DOI 10.1007/BF00534873
- A. N. Borodin, Stochastic processes, Birkhäuser Verlag, Cham, Switzerland, 2017.
- A. N. Borodin, Distribution of the supremum of increments of Brownian local time, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 142 (1985), 6–24, 195 (Russian). Problems of the theory of probability distributions, IX. MR 788182
- Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion—facts and formulae, Probability and its Applications, Birkhäuser Verlag, Basel, 1996. MR 1477407, DOI 10.1007/978-3-0348-7652-0
- Krzysztof Burdzy and Zhen-Qing Chen, Local time flow related to skew Brownian motion, Ann. Probab. 29 (2001), no. 4, 1693–1715. MR 1880238, DOI 10.1214/aop/1015345768
- E. Csáki, M. Csörgö, A. Földes, and P. Révész, Limit theorems for local and occupation times of random walks and Brownian motion on a spider, arXiv:1609.0870v2 (2017).
- E. Csáki and A. Földes, How small are the increments of the local time of a Wiener process?, Ann. Probab. 14 (1986), no. 2, 533–546. MR 832022
- Endre Csáki, Antónia Földes, and Paavo Salminen, On the joint distribution of the maximum and its location for a linear diffusion, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, 179–194 (English, with French summary). MR 891709
- H. van Haeringen and L. P. Kok, Table errata: Tables of integral transforms, Vol. II [McGraw-Hill, New York, 1954; MR 16, 468] by A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Math. Comp. 41 (1983), no. 164, 779–780. MR 717725, DOI 10.1090/S0025-5718-1983-0717725-9
- J. M. Harrison and L. A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), no. 2, 309–313. MR 606993
- K. Itô and H. P. McKean Jr., Brownian motions on a half line, Illinois J. Math. 7 (1963), 181–231. MR 154338
- Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
- Antoine Lejay, On the constructions of the skew Brownian motion, Probab. Surv. 3 (2006), 413–466. MR 2280299, DOI 10.1214/154957807000000013
- Antoine Lejay, Ernesto Mordecki, and Soledad Torres, Is a Brownian motion skew?, Scand. J. Stat. 41 (2014), no. 2, 346–364. MR 3207175, DOI 10.1111/sjos.12033
- Antoine Lejay and Géraldine Pichot, Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps, J. Comput. Phys. 231 (2012), no. 21, 7299–7314. MR 2969713, DOI 10.1016/j.jcp.2012.07.011
- Paul McGill, Markov properties of diffusion local time: a martingale approach, Adv. in Appl. Probab. 14 (1982), no. 4, 789–810. MR 677557, DOI 10.2307/1427024
- Jorge M. Ramirez, Enrique A. Thomann, and Edward C. Waymire, Advection-dispersion across interfaces, Statist. Sci. 28 (2013), no. 4, 487–509. MR 3161584, DOI 10.1214/13-STS442
- Jorge M. Ramirez, Enirque A. Thomann, and Edward C. Waymire, Continuity of local time: an applied perspective, The fascination of probability, statistics and their applications, Springer, Cham, 2016, pp. 191–207. MR 3495685
- Daniel Ray, Sojourn times of diffusion processes, Illinois J. Math. 7 (1963), 615–630. MR 156383
- L. C. G. Rogers and David Williams, Diffusions, Markov processes, and martingales. Vol. 2, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. Itô calculus. MR 921238
- Damiano Rossello, Arbitrage in skew Brownian motion models, Insurance Math. Econom. 50 (2012), no. 1, 50–56. MR 2879024, DOI 10.1016/j.insmatheco.2011.10.004
- Zhan Shi and Bálint Tóth, Favourite sites of simple random walk, Period. Math. Hungar. 41 (2000), no. 1-2, 237–249. Endre Csáki 65. MR 1812809, DOI 10.1023/A:1010389026544
- H. F. Trotter, A property of Brownian motion paths, Illinois J. Math. 2 (1958), 425–433. MR 96311
- J. B. Walsh, A diffusion with a discontinuous local time, Astérisque 52-53 (1978), 37–46.
- J. B. Walsh, Downcrossings and the Markov property of local time, Astérisque 52-53 (1978), 89–115.
Additional Information
- Andrei Borodin
- Affiliation: National Research University, Higher School of Economics, Campus Saint Petersburg, Saint Petersburg, Russia
- MR Author ID: 190985
- Email: borodin@pdmi.ras.ru
- Paavo Salminen
- Affiliation: Faculty of Science and Engineering, Abo Akademi, Turku, Finland
- MR Author ID: 153560
- Email: paavo.salminen@abo.fi
- Received by editor(s): October 18, 2018
- Received by editor(s) in revised form: February 5, 2019
- Published electronically: June 3, 2019
- Additional Notes: This research was partially supported by a grant from Magnus Ehrnrooths stiftelse, Finland, and grant SPbU-DGF 6.65.37.2017.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3597-3618
- MSC (2010): Primary 60J65, 60J60, 60J55
- DOI: https://doi.org/10.1090/tran/7852
- MathSciNet review: 3988620