Dependent Default Modeling through Multivariate Generalized Cox Processes

TL;DR

This paper introduces a flexible multivariate default modeling framework that captures complex dependence structures, including simultaneous defaults, using generalized Cox processes with non-absolutely continuous intensities.

Contribution

It extends classical Cox processes by incorporating both common and idiosyncratic shocks with non-continuous intensities, unifying various default models under a single framework.

Findings

Derived closed-form joint survival probabilities.

Demonstrated model flexibility with Lévy subordinators and shot-noise processes.

Unified gradual and abrupt default risk sources.

Abstract

We propose a multivariate framework for modeling dependent default times that extends the classical Cox process by incorporating both common and idiosyncratic shocks. Our construction uses c\`adl\`ag, increasing processes to model cumulative intensities, relaxing the requirement of absolutely continuous compensators. Analytical tractability is preserved through the multiplicative decomposition of Az\'ema supermartingales under assumptions that guarantee deterministic compensators. The framework captures a wide range of dependence structures and allows for both simultaneous and non-simultaneous defaults. We derive closed-form expressions for joint survival probabilities and illustrate the flexibility of the model through examples based on L\'evy subordinators, compound Poisson processes, and shot-noise processes, encompassing several well-known models from the literature as special cases. Finally, we show how the framework can be extended to incorporate stochastic continuous components, thereby unifying gradual and abrupt sources of default risk.