  Total value adjustment (XVA) is the change in value to be added to the price
of a derivative to account for the bilateral default risk and the funding
costs. In this paper, we compute such a premium for American basket derivatives
whose payoff depends on multiple underlyings. In particular, in our model,
those underlying are supposed to follow the multidimensional Black-Scholes
stochastic model. In order to determine the XVA, we follow the approach
introduced by Burgard and Kjaer \cite{burgard2010pde} and afterward applied by
Arregui et al. \cite{arregui2017pde,arregui2019monte} for the one-dimensional
American derivatives. The evaluation of the XVA for basket derivatives is
particularly challenging as the presence of several underlings leads to a
high-dimensional control problem. We tackle such an obstacle by resorting to
Gaussian Process Regression, a machine learning technique that allows one to
address the curse of dimensionality effectively. Moreover, the use of numerical
techniques, such as control variates, turns out to be a powerful tool to
improve the accuracy of the proposed methods. The paper includes the results of
several numerical experiments that confirm the goodness of the proposed
methodologies.
