Abstract
In this note a simple proof of the equivalence of the predictable representation property of a martingale with respect to a filtration associated with an orthogonal martingale measure and the extremality of the underlying probability measure P is given. The representation property enables us to characterize all measures which are locally absolutely continuous with respect to P. We apply this to superprocesses and remark on a related property of the excursion filtration of the Brownian motion.
Keywords
- Predictable Representation
- Orthogonal Martingale Measures
- Superprocesses
- Absolute Continuity
Supported by an EC-Fellowship under Contract No. ERBCHBICT930682 and a DFG-Fellowship.
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
[D] D. A. Dawson, Measure-valued Markov processes. In: P.L. Hennequin (ed.), Ecole d’Eté de Probabilité de Saint Flour XXI 1991, L.N.M. 1541. Springer, Berlin (1993).
[EP1] S.N. Evans, E.A. Perkins. Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 (1), 120–168 (1994).
[EP2] S.N. Evans, E.A. Perkins. Explicit stochastic integral representation for historical functionals. Preprint (1994).
[JS] J. Jacod, A.N. Shirayaev. Limit theorems for stochastic processes. Springer, Berlin (1987).
[JY] J. Jacod, M. Yor. Étude des solutions extrémales et représentation intégrale des solutions pour certains problàmes de martingales. Probab. Theory Relat. Fields 38, 83–125 (1977).
[P] E.A. Perkins. On the martingale problem for interactive measure-valued branching diffusions. To appear in Mem. Amer. Math. Soc.
[Pr] P. Protter. Stochastic Integration and Differential Equation. Springer, Berlin (1990).
[RW] L.C.G. Rogers, J.B. Walsh. Local time and stochastic area integrals. The Annals of Probability 19, 457–482 (1991).
[W] J.B. Walsh. An introduction to stochastic partial differential equation. In: P.L. Hennequin (ed.), Ecole d’Eté de Probabilité de Saint Flour XIV 1984, L.N.M. 1180, 265–439. Springer, Berlin (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag
About this paper
Cite this paper
Overbeck, L. (1995). On the predictable representation property for superprocesses. In: Azéma, J., Emery, M., Meyer, P.A., Yor, M. (eds) Séminaire de Probabilités XXIX. Lecture Notes in Mathematics, vol 1613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094203
Download citation
DOI: https://doi.org/10.1007/BFb0094203
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60219-4
Online ISBN: 978-3-540-44744-3
eBook Packages: Springer Book Archive
