zbMATH — the first resource for mathematics

An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers. (English) Zbl 1335.60151
Summary: 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.

60J65 Brownian motion
60J35 Transition functions, generators and resolvents
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C20 Probabilistic models, generic numerical methods in probability and statistics
68U20 Simulation (MSC2010)
Full Text: DOI arXiv
[1] T. Appuhamillage and D. Sheldon, First passage time of skew Brownian motion, J. Appl. Probab. 49 (2012), 3, 685-696. · Zbl 1266.60140
[2] R. Atar and A. Budhiraja, On the multi-dimensional skew Brownian motion, Stochastic Process. Appl. 125 (2015), 5, 1911-1925. · Zbl 1328.60133
[3] A.-N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, Probab. Appl., Birkhäuser, Basel, 2002. · Zbl 1012.60003
[4] D. Dereudre, S. Mazzonetto and S. Roelly, Exact simulation of Brownian diffusions with drift with several jumps, in progress. · Zbl 1370.60113
[5] P. Étoré, Approximation of one-dimensional diffusion processes with discontinuous coefficients and applications to simulation, Ph.D. thesis, University of Nancy, 2006.
[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. · Zbl 1269.65007
[7] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., De Gruyter, Berlin, 2010.
[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. · Zbl 0653.35034
[9] J.-M. Harrison and L.-A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), 2, 309-313. · Zbl 0462.60076
[10] K. Itō and H.-P. McKean, Diffusion Processes and Their Sample Paths, Academic Press, New York, 1965. · Zbl 0127.09503
[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. · Zbl 0551.60059
[12] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv. 3 (2006), 413-466. · Zbl 1189.60145
[13] A. Lejay, L. Lenôtre and G. Pichot, One-dimensional skew diffusions: Explicit expressions of densities and resolvent kernel, preprint 2015, https://hal.inria.fr/hal-01194187v2.
[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. · Zbl 1094.60056
[15] Y. Ouknine, Le “Skew-Brownian motion” et les processus qui en dérivent, Teor. Veroyatnost. i Primenen. 35 (1990), 1, 173-179. · Zbl 0696.60080
[16] Y. Ouknine, F. Russo and G. Trutnau, On countably skewed Brownian motion with accumulation point, Electron. J. Probab. 20 (2015), 82, 1-27. · Zbl 1327.31023
[17] M.-I. Portenko, Diffusion processes with a generalized drift coefficient, Teor. Veroyatnost. i Primenen. 24 (1979), 1, 62-77. · Zbl 0396.60071
[18] M.-I. Portenko, Generalized Diffusion Processes, Transl. Math. Monogr. 83, American Mathematical Society, Providence, 1990.
[19] J.-M. Ramirez, Multi-skewed Brownian motion and diffusion in layered media, Proc. Amer. Math. Soc. 139 (2011), 10, 3739-3752. · Zbl 1231.60084
[20] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts Appl. Math., Springer, New York, 2006. · Zbl 0917.35001
[21] S.-M. Ross, Simulation, Academic Press, New York, 2013. · Zbl 1255.65001
[22] D. Veestraeten, The conditional probability density function for a reflected Brownian motion, Comput. Econ. 24 (2004), 2, 185-207. · Zbl 1067.60082
[23] J. von Neumann, Various techniques used in connection with random digits. Monte Carlo methods, Natl. Bureau Standards 12 (1951), 36-38.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.