  The X-valuation adjustment (XVA) problem, which is a recent topic in
mathematical finance, is considered and analyzed. First, the basic properties
of backward stochastic differential equations (BSDEs) with a random horizon in
a progressively enlarged filtration are reviewed. Next, the pricing/hedging
problem for defaultable over-the-counter (OTC) derivative securities is
described using such BSDEs. An explicit sufficient condition is given to ensure
the non-existence of an arbitrage opportunity for both the seller and buyer of
the derivative securities. Furthermore, an explicit pricing formula is
presented in which XVA is interpreted as approximated correction terms of the
theoretical fair price.
