Abstract
We give sufficient conditions under which a continuous local martingale remains a continuous local martingale under a simultaneous initial expansion of the filtration of σ-fields and change to an equivalent probability law. In particular this gives a method for a Brownian motion to remain a Brownian motion under such a double transformation. The classical example of K. Itô is treated in detail.
Key words and phrases
- Brownian motion
- local martingale
- semimartingale
- initial expansion of filtrations
- Girsanov's theorem
Supported in part by NSF Grant # 8500997
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C. Dellacherie, P.-A. Meyer: Probabilities and Potential B, North-Holland, 1982.
K. Itô: Extension of Stochastic Integrals. Proc. of the Int. Symp. on Stochastic Differential Equations, 95–109. Wiley, New York.
J. Jacod: Grossissement Initial, Hypothèse (H') et Théorème de Girsanov. Springer Lecture Notes in Math. 1118, 15–35, 1985.
J. Jacod, P. Protter: Time Reversal on Lévy Processes. Annals of Probability 16, 620–641, 1988.
T. Jeulin, M. Yor: Inégalité de Hardy, semimartingales, et faux-amis. Springer Lecture Notes in Math. 721, 332–359, 1979.
N. Kazamaki: On a Problem of Girsanov. Tôhoku Math. Journal 29, 597–600, 1977.
M. Yor: Grossissement de Filtrations et Absolue Continuité de Noyaux. Springer Lecture Notes in Math. 1118, 6–14, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Protter, P. (1989). A connection between the expansion of filtrations and Girsanov's theorem. In: Da Prato, G., Tubaro, L. (eds) Stochastic Partial Differential Equations and Applications II. Lecture Notes in Mathematics, vol 1390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083949
Download citation
DOI: https://doi.org/10.1007/BFb0083949
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51510-4
Online ISBN: 978-3-540-48200-0
eBook Packages: Springer Book Archive
