Abstract
For a Banach-valued martingaleX, we define anL 1-valued measureJ X on an algebra of stochastic intervals which generates the optional σ-algebraO. We discuss conditions for when the measure has a countably additive extension toO, that is, for whenX isO-summable. For a process of integrable variationV, we define another countably additive measureI V onO. The existence of these measures allows for the definition of stochastic integrals of optional processes with respect to these Banach-valued processesX andV.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brooks, J. K., and Dinculeanu, N. (1986)H 1 andBMP spaces of abstract Martingales. InSeminar on Stochastic Processes 1985 Birkhäuser, Boston.
Brooks, J. K., and Dinculeanu, N. (1988). Regularity and the Doob-Meyer Decomposition of Abstract Quasimartingales. InSeminar on Stochastic Processes 1987 Birkhäuser, Boston.
Brooks, J. K. and Dinculeanu, N. (1991). Stochastic integration in Banach spaces. InSeminar on Stochastic Processes 1990 Birkhäuser, Boston.
Brooks, J. K., and Neal, D. A vector measure approach to the optional stochastic integral.Ulam Quarterly 1, 17–25.
Déllacherie, C., Meyer, P. (1982).Probabilities and Potential. North-Holland, New York.
Dinculeanu, N. (1988). Vector valued stochastic processes III: Projections and dual projections. InSeminar on Stochastic Processes 1987. Birkhäuser, Boston.
Kussmaul, A. U. (1987).Stochastic Integration and Generalized Martingales. Pitman, London.
Kwapien, S. (1974). On Banach spaces containingc 0.Studia Math. 5, 187–188.
Protter, P. (1990).Stochastic Integration and Differential Equations. Springer-Verlag, New York.
Schwartz, L. (1984).Semimartingales and their Stochastic Calculus on Manifolds. Les Presses de l'Université de Montréal, Montréal.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neal, D. Characterization ofO-summable processes. J Theor Probab 5, 585–596 (1992). https://doi.org/10.1007/BF01060438
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01060438