Mild to Classical Solutions for XVA Equations under Stochastic Volatility

.We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.