🤖 AI Summary
This paper addresses the lack of rigorous mathematical foundations for Local Stochastic Intensity (LSI) models in credit risk modeling. Specifically, it resolves, for the first time, the Markovian projection inverse problem for pure-jump processes: reconstructing the original jump dynamics reversibly from knowledge of only the one-dimensional marginal distribution. Methodologically, the approach integrates Itô semimartingale theory, decomposition of random measures, Lévy-driven local time change, and inversion of projection operators to achieve exact marginal distribution matching. The key contribution is the construction of the first explicitly calibratable LSI model, thereby providing theoretical grounding for jump-type Local Stochastic Volatility (LSV) frameworks in credit risk. Empirically, the model significantly improves the fit of default probability curves and constitutes the first analytically tractable framework for high-dimensional jump 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 simpler 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.