Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem

TL;DR

This paper extends the Jeulin--Yor theorem to develop a new filtration expansion approach for computing the intensity process of a stopping time, with applications in credit risk and martingale characterization.

Contribution

It introduces a novel methodology for calculating the intensity process of a stopping time using filtration expansion techniques, extending classical results.

Findings

Extended Jeulin--Yor theorem for general filtration expansions

Derived a characterization of martingales under local jumping filtrations

Proposed a new approach for intensity process computation in credit risk

Abstract

Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov process with possible jumps and let $\tau$ be its first hitting time of a Borel subset of the state space. Suppose $X$ is sampled at random times and suppose also that $X$ has not hit the Borel set by time $t$. What is the intensity process of $\tau$ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin--Yor [Lecture Notes in Math. 649 (1978) 78--97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21--35] under local jumping filtration is derived.