We extend the valuation of contingent claims in presence of default, collateral and
funding to a random functional setting and characterise 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 characterise 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.