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Characterization ofO-summable processes

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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.

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  1. 1.

    Brooks, J. K., and Dinculeanu, N. (1986)H 1 andBMP spaces of abstract Martingales. InSeminar on Stochastic Processes 1985 Birkhäuser, Boston.

  2. 2.

    Brooks, J. K., and Dinculeanu, N. (1988). Regularity and the Doob-Meyer Decomposition of Abstract Quasimartingales. InSeminar on Stochastic Processes 1987 Birkhäuser, Boston.

  3. 3.

    Brooks, J. K. and Dinculeanu, N. (1991). Stochastic integration in Banach spaces. InSeminar on Stochastic Processes 1990 Birkhäuser, Boston.

  4. 4.

    Brooks, J. K., and Neal, D. A vector measure approach to the optional stochastic integral.Ulam Quarterly 1, 17–25.

  5. 5.

    Déllacherie, C., Meyer, P. (1982).Probabilities and Potential. North-Holland, New York.

  6. 6.

    Dinculeanu, N. (1988). Vector valued stochastic processes III: Projections and dual projections. InSeminar on Stochastic Processes 1987. Birkhäuser, Boston.

  7. 7.

    Kussmaul, A. U. (1987).Stochastic Integration and Generalized Martingales. Pitman, London.

  8. 8.

    Kwapien, S. (1974). On Banach spaces containingc 0.Studia Math. 5, 187–188.

  9. 9.

    Protter, P. (1990).Stochastic Integration and Differential Equations. Springer-Verlag, New York.

  10. 10.

    Schwartz, L. (1984).Semimartingales and their Stochastic Calculus on Manifolds. Les Presses de l'Université de Montréal, Montréal.

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Neal, D. Characterization ofO-summable processes. J Theor Probab 5, 585–596 (1992). https://doi.org/10.1007/BF01060438

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Key words

  • Stochastic integral
  • vector-valued measures
  • summable andO-summable processes