Representation for martingales living after a random time with applications

TL;DR

This paper develops explicit representations for martingales after a random time in enlarged filtrations, with applications to credit risk and life insurance models, including jump-diffusion and discrete-time cases.

Contribution

It extends existing martingale representation results to the case when martingales live after a random time T, providing explicit parametrizations of deflators in enlarged filtrations.

Findings

Explicit G-local martingale representations in terms of F-martingales.

Application to jump-diffusion and discrete-time models.

Complete characterization of deflators under enlarged filtration.

Abstract

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