Invariance properties in the dynamic gaussian copula model

We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Cr{\'e}pey, Jeanblanc, and Wu (2013) are invariance times in the sense of Cr{\'e}pey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times. These properties are in line with the wrong-way risk feature of counterparty risk embedded in credit derivatives, i.e. the adverse dependence between the default risk of a counterparty and an underlying credit derivative exposure.