Abstract
In a previous work, we have defined a Tanaka’s SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain these solutions as limits of discrete models.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.S. Cherny, A.N. Shiryaev, M. Yor, Limit behaviour of the “horizontal-vertical” random walk and some extensions of the Donsker-Prokhorov invariance principle. Teor. Veroyatnost. i Primenen 47(3) 498–517 (2002)
N. Enriquez, Y. Kifer, Markov chains on graphs and Brownian motion. J. Theoret. Probab. 14(2) 495–510 (2001)
S. Ethier, T. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, (Wiley, New York, 1986)
J.M. Harrison, L.A. Shepp, On skew Brownian motion. Ann. Probab. 9(2) 309–313 (1981)
H. Hajri, Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16, 1563–1599 (2011)
Y. Le Jan, O. Raimond, Flows, coalescence and noise. Ann. Probab. 32(2), 1247–1315 (2004)
Y. Le Jan, O. Raimond, Flows associated to Tanaka’s SDE. ALEA Lat. Am. J. Probab. Math. Stat. 1, 21–34 (2006)
I.I. Nishchenko, Discrete time approximation of coalescing stochastic flows on the real line. Theory Stoch. Proc. 17(33)(1), 70–78 (2011)
P. Révész, Random Walk in Random and Non-random Environments. 2nd ed. (World Scientific Publishing, 2005)
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. (Grundlehren der Mathematischen Wissenschaften, 1999)
S. Watanabe, The stochastic flow and the noise associated to Tanaka stochastic differential equation. Ukraïn. Mat. Zh. 52(9), 1176–1193 (2000)
Acknowledgements
I sincerely thank Yves Le Jan, Olivier Raimond and Sophie Lemaire for very useful discussions. I am also grateful to the referee for his helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hajri, H. (2012). Discrete Approximations to Solution Flows of Tanaka’s SDE Related to Walsh Brownian Motion. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIV. Lecture Notes in Mathematics(), vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27461-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-27461-9_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27460-2
Online ISBN: 978-3-642-27461-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)