  We study conditions for existence, uniqueness and invariance of the
comprehensive nonlinear valuation equations first introduced in Pallavicini et
al (2011). These equations take the form of semilinear 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 viscosity and 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 semilinear PDE or FBSDEs and
their classical or viscosity 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 Pallavicini et al (2011), Pallavicini et al (2012),
Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).
