Inverting the Markovian projection for pure jump processes
Martin Larsson, Shukun Long
TL;DR
This work addresses inverting Markovian projections for pure jump Itô semimartingales by constructing richer processes whose one-dimensional marginals match those of a given Markovian projection. The authors develop existence results using the Cox construction and fixed-point methods, first for counting processes (jump size 1) and then for general discrete jump sizes, establishing the existence of a doubly stochastic Poisson/compound Poisson projection $\widehat{X}$ that preserves marginals. They show that uniqueness in law generally fails but provide sufficient conditions (immersion and positive conditional survival) under which the inversion is unique, with a parallel fixed-point framework solving for conditional intensities. The generalization to arbitrary discrete jump sizes relies on Schauder fixed points and Hölder regularity, yielding a robust framework for calibrating local stochastic intensity models in credit risk, analogous to LSV calibrations in equity settings.
Abstract
Markovian projections arise in problems where we aim to mimic the one-dimensional marginal laws of an Itô semimartingale by using another Itô 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.