We study conditions for existence, uniqueness and invariance of the comprehensive
nonlinear valuation equations first introduced in Pallavicini et al (2011)
[11]. These equations take the form of semi-linear PDEs and Forward-Backward
Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions
allowing us to extend valuation to credit risk and default closeout, including
collateral margining with possible re-hypothecation, and treasury funding costs, we
show how such cash flows, when present-valued in an arbitrage free setting, lead
to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence
and uniqueness of such solutions in a classical sense, discussing the role of the
hedging strategy. We show an invariance theorem stating that even though we start
from a risk-neutral valuation approach based on a locally risk-free bank account
growing at a risk-free rate, our final valuation equations do not depend on the risk
free rate. Indeed, our final semi-linear PDE or FBSDEs and their classical solutions
depend only on contractual, market or treasury rates and we do not need to proxy
the risk free rate with a real market rate, since it acts as an instrumental variable. The
equations derivations, their numerical solutions, the related XVA valuation adjustments
with their overlap, and the invariance result had been analyzed numerically
and extended to central clearing and multiple discount curves in a number of previous
works, including [11], [12], [10], [6] and [4].