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.

翻译：在本文中,我们研究了可用于模拟价值调整的局部差分方程(PDEs),现在一般称为xVA的不同价值调整被添加到无风险的金融衍生物价值中,而PDE方法便于其纳入。本文的目的是展示如何在黑雪圈环境中分析解决PDE,以获得新的半封闭公式,与广泛使用的Monte-Carlo模拟和PDE的数字解决方案进行比较。以过去价值为抵押品的例子将引起人们的兴趣。