Abstract
We will show, using the skew product representation and the corresponding result for the radial process, that Shiga–Watanabe’s time inversion property of index α, α positive, holds for all α-self-similar, rotation invariant diffusion processes on R { d}, d ≥ 2, starting at 0.
AMS Classification: 60G18, 60J60, 60J25
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
L. Gallardo, M. Yor, Some new examples of Markov processes which enjoy the time-inversion property. Probab. Theor. Relat. Field. 132(1), 150–162 (2005)
S.E. Graversen, J. Vuolle-Apiala, α-self-similar Markov processes. Probab. Theor. Relat. Field. 71(1), 149–158 (1986)
S.E. Graversen, J. Vuolle-Apiala, On Paul Levy’s arc sine law and Shiga-Watanabe’s time inversion result. Probab. Math. Statist. 20(1), 63–73 (2000)
K. Ito, H.P. McKean Jr., Diffusion Processes and Their Sample Paths (Springer, Berlin, 1974)
J.W. Lamperti, Semi-stable Markov processes I. Z. Wahrschein. verw. Gebiete 22, 205–225 (1972)
S. Lawi, Towards a characterization of Markov processes enjoying the time-inversion property. J. Theoret. Probab. 21(1), 144–168 (2008)
D. Revuz, M. Yor, Continuous martingales and Brownian motion (Springer, Berlin, 1991)
T. Shiga, S. Watanabe, Bessel diffusions as one parameter family of diffusion processes. Z. Wahrschein. verw. Gebiete 27, 37–46 (1973)
J. Vuolle-Apiala, On certain extensions of a rotation invariant Markov process. J. Theoret. Probab. 16(4), 957–969 (2003)
J. Vuolle-Apiala, Time inversion result for self-similar diffusions on finite number of rays, in Contributions to Management Science, Mathematics and Modelling, Acta Wasaensia 122, 253–260 (University of Vaasa, 2004)
J. Vuolle-Apiala, Shiga-Watanabe’s time inversion property for self-similar diffusion processes. Stat. Probab. Lett. 77(9), 920–924 (2007)
J. Vuolle-Apiala, S.E. Graversen, Duality theory for self-similar processes. Ann. Inst. Henri Poincaré 22, 323–332 (1986)
S. Watanabe, On time inversion of one-dimensional diffusion process. Z. Wahrschein. verw. Gebiete 31, 115–124 (1975)
Acknowledgements
The author wants to thank an anonymous referee for useful comments and suggestions which considerably improved this paper.
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
Vuolle-Apiala, J. (2012). Time Inversion Property for Rotation Invariant Self-similar Diffusion Processes. 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_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-27461-9_13
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)