$\natural$-model with jumps

TL;DR

This paper extends the $ atural$-model for random times to include jump parameters, providing new formulas for filtration enlargement and derivatives of conditional distributions, advancing the mathematical understanding of default modeling.

Contribution

It introduces the $ atural$-model with jump parameters and derives the associated filtration enlargement and distribution derivatives, filling a gap in existing continuous-parameter models.

Findings

Established the $ atural$-model with jump parameters.

Derived the enlargement of filtration formula for jump models.

Computed derivatives of conditional distribution functions.

Abstract

We consider the so-called $\natural$-model. It is an one-default model which gives the conditional law of a random time with respect to a reference filtration. This model has been studied in the case where the parameters are continuous. In this paper we will establish the $\natural$-model in the case of jump parameters. We then prove the corresponding enlargement of filtration formula and we compute the derivative of the conditional distribution functions of the random time.