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An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers

  • David Dereudre , Sara Mazzonetto EMAIL logo and Sylvie Roelly


In this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover, we propose a rejection sampling method to simulate this density in an exact way.

Funding source: Deutsch-Französische Hochschule – Université Franco-Allemande (DFH-UFA)

Funding source: Stochastic Analysis with Applications in Biology, Finance and Physics

Award Identifier / Grant number: RTG 1845

The authors would like to thank Patrick Cattiaux, Markus Klein, Andrey Pilipenko and Lionel Lenôtre for the interesting discussions.


1 T. Appuhamillage and D. Sheldon, First passage time of skew Brownian motion, J. Appl. Probab. 49 (2012), 3, 685–696. 10.1239/jap/1346955326Search in Google Scholar

2 R. Atar and A. Budhiraja, On the multi-dimensional skew Brownian motion, Stochastic Process. Appl. 125 (2015), 5, 1911–1925. 10.1016/ in Google Scholar

3 A.-N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, Probab. Appl., Birkhäuser, Basel, 2002. 10.1007/978-3-0348-8163-0Search in Google Scholar

4 D. Dereudre, S. Mazzonetto and S. Roelly, Exact simulation of Brownian diffusions with drift with several jumps, in progress. Search in Google Scholar

5 P. Étoré, Approximation of one-dimensional diffusion processes with discontinuous coefficients and applications to simulation, Ph.D. thesis, University of Nancy, 2006. 10.1214/EJP.v11-311Search in Google Scholar

6 P. Étoré and M. Martinez, Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process, Monte Carlo Methods Appl. 19 (2013), 1, 41–71. 10.1515/mcma-2013-0002Search in Google Scholar

7 M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., De Gruyter, Berlin, 2010. 10.1515/9783110218091Search in Google Scholar

8 B. Gaveau, M. Okada and T. Okada, Second order differential operators and Dirichlet integrals with singular coefficients I. Functional calculus of one-dimensional operators, Tohoku Math. J. (2) 39 (1987), 4, 465–504. 10.2748/tmj/1178228238Search in Google Scholar

9 J.-M. Harrison and L.-A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), 2, 309–313. 10.1214/aop/1176994472Search in Google Scholar

10 K. Itō and H.-P. McKean, Diffusion Processes and Their Sample Paths, Academic Press, New York, 1965. Search in Google Scholar

11 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. 1095, Springer, Berlin (1984), 51–82. 10.1007/BFb0099122Search in Google Scholar

12 A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv. 3 (2006), 413–466. 10.1214/154957807000000013Search in Google Scholar

13 A. Lejay, L. Lenôtre and G. Pichot, One-dimensional skew diffusions: Explicit expressions of densities and resolvent kernel, preprint 2015, Search in Google Scholar

14 A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, Ann. Appl. Probab. 16 (2006), 1, 107–139. 10.1214/105051605000000656Search in Google Scholar

15 Y. Ouknine, Le “Skew-Brownian motion” et les processus qui en dérivent, Teor. Veroyatnost. i Primenen. 35 (1990), 1, 173–179. Search in Google Scholar

16 Y. Ouknine, F. Russo and G. Trutnau, On countably skewed Brownian motion with accumulation point, Electron. J. Probab. 20 (2015), 82, 1–27. 10.1214/EJP.v20-3640Search in Google Scholar

17 M.-I. Portenko, Diffusion processes with a generalized drift coefficient, Teor. Veroyatnost. i Primenen. 24 (1979), 1, 62–77. 10.1137/1124005Search in Google Scholar

18 M.-I. Portenko, Generalized Diffusion Processes, Transl. Math. Monogr. 83, American Mathematical Society, Providence, 1990. 10.1090/mmono/083Search in Google Scholar

19 J.-M. Ramirez, Multi-skewed Brownian motion and diffusion in layered media, Proc. Amer. Math. Soc. 139 (2011), 10, 3739–3752. 10.1090/S0002-9939-2011-10766-4Search in Google Scholar

20 M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts Appl. Math., Springer, New York, 2006. Search in Google Scholar

21 S.-M. Ross, Simulation, Academic Press, New York, 2013. Search in Google Scholar

22 D. Veestraeten, The conditional probability density function for a reflected Brownian motion, Comput. Econ. 24 (2004), 2, 185–207. 10.1023/B:CSEM.0000049491.13935.afSearch in Google Scholar

23 J. von Neumann, Various techniques used in connection with random digits. Monte Carlo methods, Natl. Bureau Standards 12 (1951), 36–38. Search in Google Scholar

Received: 2015-9-8
Accepted: 2016-2-2
Published Online: 2016-2-17
Published in Print: 2016-3-1

© 2016 by De Gruyter

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