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

Gapeev, Pavel V. ORCID: 0000-0002-1346-2074 and Jeanblanc, Monique (2024) On the construction of conditional probability densities in the Brownian and compound Poisson filtrations. ESAIM: Probability and Statistics, 28. 62 - 74. ISSN 1292-8100

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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.

Item Type:	Article
Official URL:	https://www.esaim-ps.org/
Additional Information:	© 2024 The Authors
Divisions:	Mathematics
Subjects:	H Social Sciences > HA Statistics Q Science > QA Mathematics
Date Deposited:	14 Dec 2023 16:57
Last Modified:	15 Nov 2024 01:48
URI:	http://eprints.lse.ac.uk/id/eprint/121059

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