On the Construction of Conditional Probability Densities in the Brownian and Compound Poisson Filtrations

¹ London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK
² Laboratoire de Mathématiques et Modélisation d’Évry (LaMME), UMR CNRS 8071; Univ Evry-Université Paris Saclay, 23 Boulevard de France, 91037 Évry cedex France

^* Corresponding author: p.v.gapeev@lse.ac.uk

Received: 29 July 2022
Accepted: 13 December 2023

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.