Inverting the Markovian projection for pure jump processes

TL;DR

This paper develops a method to invert Markovian projections for pure jump processes, enabling the construction of calibrated local stochastic intensity models for credit risk, analogous to complex volatility models in equity markets.

Contribution

It introduces an inversion technique for Markovian projections of pure jump processes, facilitating the creation of local stochastic intensity models for credit risk applications.

Findings

Provides a new inversion method for pure jump processes

Enables calibration of local stochastic intensity models

Facilitates applications in credit risk modeling

Abstract

Markovian projections arise in problems where we aim to mimic the one-dimensional marginal laws of an It\^o semimartingale by using another It\^o process with Markovian dynamics. In applications, Markovian projections are useful in calibrating jump-diffusion models with both local and stochastic features, leading to the study of the inversion problems. In this paper, we invert the Markovian projections for pure jump processes, which can be used to construct calibrated local stochastic intensity (LSI) models for credit risk applications. Such models are jump process analogues of the notoriously hard to construct local stochastic volatility (LSV) models used in equity modeling.