Invariance Times Transfer Properties

TL;DR

This paper studies invariance times in stochastic processes, establishing a comprehensive framework relating the calculus of semimartingales before and after such times, with applications to credit risk modeling.

Contribution

It develops a detailed dictionary linking semimartingale properties across original and reduced filtrations at invariance times, expanding theoretical understanding in stochastic calculus.

Findings

Established relations for conditional expectations and martingales across filtrations.

Derived semimartingale characteristics and transition semigroups at invariance times.

Applied results to backward stochastic differential equations in credit risk context.

Abstract

Invariance times are stopping times $\tau$ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $\tau$ , are local martingales with respect to the original model filtration and probability measure. They arise naturally for modeling the default time of a dealer bank, in the mathematical finance context of counterparty credit risk. Assuming an invariance time endowed with an intensity and a positive Az{\'e}ma supermartingale, this work establishes a dictionary relating the semimartingale calculi in the original and reduced stochastic bases, regarding in particular conditional expectations, martingales, stochastic integrals, random measure stochastic integrals, martingale representation properties, semimartingale characteristics, Markov properties, transition semigroups and infinitesimal generators, and solutions of backward stochastic differential equations.