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Characterization ofO-summable processes

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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.

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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