On Jump Measures of Optional Processes with Regulated Trajectories

TL;DR

This paper develops a new representation for the jumps of optional processes with regulated trajectories, facilitating analysis and integration with respect to jump measures, especially in the context of enlarged filtrations in finance.

Contribution

It introduces a novel iterative representation technique for jumps of optional processes with regulated trajectories, extending previous methods for càdlàg processes.

Findings

Representation of jump sets for optional processes with regulated trajectories

Application to integration with respect to jump measures

Framework for enlarged filtrations in financial mathematics

Abstract

Starting from an iterative and hence numerically easily implementable representation of the thin set of jumps of a c\`{a}dl\`{a}g adapted stochastic process $X$ (including a few applications to the integration with respect to the jump measure of $X$), we develop similar representation techniques to describe the set of jumps of optional processes with regulated trajectories and introduce their induced jump measures with a view towards the framework of enlarged filtration in financial mathematics.