  In this paper we study partial differential equations (PDEs) that can be used
to model value adjustments. Different value adjustments denoted generally as
xVA are nowadays added to the risk-free financial derivative values and the PDE
approach allows their easy incorporation. The aim of this paper is to show how
to solve the PDE analytically in the Black-Scholes setting to get new
semi-closed formulas that we compare to the widely used Monte-Carlo simulations
and to the numerical solutions of the PDE. Particular example of collateral
taken as the values from the past will be of interest.
