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.KeywordsXVAvaluationcollateralfunding costsdefault timestochastic volatilitystochastic differential equationmild solutionsemilinear parabolic PDEMSC codes91G2091G8060G4060H2060H3035K58