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On the construction of conditional probability densities in the Brownian and compound Poisson filtrations

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  • Gapeev, Pavel V.
  • Jeanblanc, Monique

Abstract

In this paper, we construct supermartingales valued in [0,1] as solutions of an appropriate stochastic differential equation on a given reference filtration generated by either a Brownian motion or a compound Poisson process. Then, by means of the results contained in [M. Jeanblanc and S. Song, Stochastic Processes Appl. 121 (2011) 1389–1410], it is possible to construct an associated random time on some extended probability space admitting such a given supermartingale as conditional survival process and we shall check that this construction (with a particular choice of supermartingale) implies that Jacod’s equivalence hypothesis, that is, the existence of a family of strictly positive conditional probability densities for the random times with respect to the reference filtration, is satisfied. We use the components of the multiplicative decomposition of the constructed supermartingales to provide explicit expressions for the conditional probability densities of the random times on the Brownian and compound Poisson filtrations.

Suggested Citation

  • Gapeev, Pavel V. & Jeanblanc, Monique, 2024. "On the construction of conditional probability densities in the Brownian and compound Poisson filtrations," LSE Research Online Documents on Economics 121059, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:121059
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    Keywords

    conditional probability density process; Brownian motion; compound Poisson process; Jacod's equivalence hypothesis; Small Grant from the Suntory and Toyota International Centres for Economics and Related Disciplines (STICERD) at the LSE;
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    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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