Closed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processes

Arturo Valdivia

doi:10.1214/19-ecp221

Abstract

We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by Levy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.

Highlights

Consider an ordered series of random times τ1 ≤ ... ≤ τm accounting for the sequenced occurrence of certain events
We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process
We translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials

Summary

Introduction

Consider an ordered series of random times τ1 ≤ ... ≤ τm accounting for the sequenced occurrence of certain events. Closed-form formulas for the distribution of the jumps of a DSP processes problem from two different approaches We relate this problem to the computation of the first n derivatives of the Hazard process cumulant generating function. We compute the aforementioned distributions directly, by means of the Malliavin calculus For this approach we consider a strictly positive purejump Lévy process (Lt)t≥0 with Lévy measure ν, and having moments of all orders —see [1, 17] and [7] for a general exposition about Lévy processes and Malliavin calculus. Assume further that F is given by the natural filtration generated by the driving Lévy process (Lt)t≥0 In this setting, we have the following result. Let us remark that even though our study is motivated by the valuation of defaultable claims, our results can potentially be used in other areas; see for instance [2, 14, 21] and references therein

Examples

Proofs

Preliminaries on Malliavin calculus via chaos expansions

A recursive formula