Abstract
We consider an R d-valued continous semimartingale (X t ) t∃[0, T] , the space of processes G p = {θ · X | θ · X is a semimartingale in S p} and the space of their terminal values G P T . We give necessary and sufficient conditions for completeness of G P in the norm ∥(θ · X)*∥ p and closedness of G P T in L p. These results are related to some problems in mathematical finance and have been given for p=2 in [DMSSS].
Supported by “Fonds zur Förderung der wissenschaftlichen Forschung in Österreich”, Project Nr. P11544
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
[AS] Ansel J.P. and Stricker C, ‘Lois de martingale, densités et décomposition de Föllmer-Schweizer', Annales de l'Institut Henri Poincaré 28, 375–392 (1992).
[DM] Doleans-Dade C. and Meyer P.A., ‘Inégalités de normes avec poids', Séminaire de Probabilités XIII, Lecture Notes In Mathematics 721, 313–331 (1977).
[DMSSS] Delbaen F., Monat P., Schachermayer W., Schweizer M., Stricker C., ‘Weighted Norm Inequalities and Closedness of a Space of Stochastic Integrals', to appear in Finance and Stochastics.
[DS1] Delbaen F., Schachermayer W., ‘The Existence of Absolutely Continuous Local Martingale Measures', Annals of Applied Probability Vol. 5, No. 4, 926–945 (1995).
[DS2] Delbaen F., Schachermayer W., ‘The Variance Optimal Martingale Measure for Continuous Processes', Bernoulli 2 (1), 81–105 (1996).
[K] Kazamaki N., ‘Exponential Continuous martingales and BMO', Lecture Notes in Mathematics 1579, Springer-Verlag 1994.
[Lb] Luenberger D. G., ‘Optimization by Vector Space Methods', Wiley and Sons 1969.
[P] Pratelli M., ‘Sur certains espaces de martingales localement de carré intégrable', Séminaire de Probabilités X, Lecture Notes In Mathematics 511, 401–413 (1975).
[RY] Revuz D., Yor M., ‘Continuous Martingales and Brownian Motion', Springer-Verlag, Berlin 1991.
[Y] Yor M., ‘Inegalités de martingales continues arrêtées à un temps quelconque', Lecture Notes In Mathematics 1118, Springer-Verlag 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag
About this paper
Cite this paper
Grandits, P., Krawczyk, L. (1998). Closedness of some spaces of stochastic integrals. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101752
Download citation
DOI: https://doi.org/10.1007/BFb0101752
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64376-0
Online ISBN: 978-3-540-69762-6
eBook Packages: Springer Book Archive