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2017 | OriginalPaper | Chapter

2. The Reflecting Brownian Motion

Author : Lorenzo Zambotti

Published in: Random Obstacle Problems

Publisher: Springer International Publishing

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Abstract

In this chapter we study SDEs in \(\mathbb{R}_{+}:= [0,+\infty [\) with reflection at 0. We give two different approaches: the classical one based on the Skorokhod Lemma, and the penalisation method. The interest of the latter lies in its applicability, in Chap. 5 below, to stochastic partial differential equations. We discuss the link between reflection and local times and we give a formula, in Lemma 2.9 below, which will be useful later, in particular in the construction of Bessel processes in Chap. 3

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Aid15.

S. Aida, Reflected rough differential equations. Stoch. Process. Appl. 125 (9), 3570–3595 (2015). doi:10.1016/j.spa.2015.03.008

Ber99.

J. Bertoin, Subordinators: examples and applications, in Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Mathematics, vol. 1717 (Springer, Berlin, 1999), pp. 1–91. doi:10.1007/978-3-540-48115-7_1

Cép98.

E. Cépa, Problème de Skorohod multivoque. Ann. Probab. 26 (2), 500–532 (1998). doi:10.1214/aop/1022855642

DGHT16.

A. Deya, M. Gubinelli, M. Hofmanova, S. Tindel, One-dimensional reflected rough differential equations. ArXiv e-prints (2016). arXiv:1610.07481

LS84.

P.-L. Lions, A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37 (4), 511–537 (1984). doi:10.1002/cpa.3160370408

RY99.

D. Revuz, M. Yor, Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. (Springer, Berlin, 1999), pp. xiv+602. doi:10.1007/978-3-662-06400-9

Sko61.

A.V. Skorohod, Stochastic equations for diffusion processes with a boundary. Teor. Verojatnost. i Primenen. 6, 287–298 (1961)MathSciNet

Sko62.

A.V. Skorohod, Stochastic equations for diffusion processes with boundaries. II. Teor. Verojatnost. i Primenen. 7, 5–25 (1962)MathSciNet

Tan79.

H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1), 163–177 (1979)MathSciNetMATH

Title: The Reflecting Brownian Motion
Author: Lorenzo Zambotti
Publisher: Springer International Publishing
Book: Random Obstacle Problems
Print ISBN: 978-3-319-52095-7

Electronic ISBN: 978-3-319-52096-4

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-52096-4_2