Representation for martingales living after a random time with applications

Abstract

Our financial setting consists of a market model with two flows of information. The smallest flow $ \mathbb{F} $ is the 'public' flow of information which is available to all agents, while the larger flow $ \mathbb{G} $ has additional information about the occurrence of a random time $ \tau $. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration $ \mathbb{G} $ is the progressive enlargement of $ \mathbb{F} $ with $ \tau $. In this framework, when $ \tau $ is a finite honest time, we describe explicitly how $ \mathbb G $-local martingales can be represented in terms of $ \mathbb{F} $-local martingales and parameters of $ \tau $. This representation complements Choulli, Daveloose and Vanmaele [12] to the case when martingales live 'after $ \tau $'. Under some mild assumptions on the pair $ (\mathbb{F}, \tau) $, we fully elaborate the application of these results to the explicit parametrization of all deflators under $ \mathbb{G} $. The results are illustrated in the case of a jump-diffusion model and a discrete-time market model.