Invariance times transfer properties

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 Azema supermartingale, this work establishes a dictionary relating the semimartingale calculi in the original and reduced stochastic bases, regarding 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.