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Multiplicative Decompositions and Frequency of Vanishing of Nonnegative Submartingales

Abstract

In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y = N + A, where the measure dA is carried by the set of zeros of Y. In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.

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Correspondence to Ashkan Nikeghbali.

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Nikeghbali, A. Multiplicative Decompositions and Frequency of Vanishing of Nonnegative Submartingales. J Theor Probab 19, 931–949 (2006). https://doi.org/10.1007/s10959-006-0035-0

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  • DOI: https://doi.org/10.1007/s10959-006-0035-0

Keywords

  • random times
  • submartingales
  • general theory of stochastic processes

2000 Mathematics Subject Classification

  • 05C38
  • 15A15
  • 05A15
  • 15A18