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Total value adjustment (XVA) is the change in value to be added to the priceof a derivative to account for the bilateral default risk and the fundingcosts. In this paper, we compute such a premium for American basket derivativeswhose payoff depends on multiple underlyings. In particular, in our model,those underlying are supposed to follow the multidimensional Black-Scholesstochastic model. In order to determine the XVA, we follow the approachintroduced by Burgard and Kjaer \cite{burgard2010pde} and afterward applied byArregui et al. \cite{arregui2017pde,arregui2019monte} for the one-dimensionalAmerican derivatives. The evaluation of the XVA for basket derivatives isparticularly challenging as the presence of several underlings leads to ahigh-dimensional control problem. We tackle such an obstacle by resorting toGaussian Process Regression, a machine learning technique that allows one toaddress the curse of dimensionality effectively. Moreover, the use of numericaltechniques, such as control variates, turns out to be a powerful tool toimprove the accuracy of the proposed methods. The paper includes the results ofseveral numerical experiments that confirm the goodness of the proposedmethodologies.

Computing XVA for American basket derivatives by Machine Learning techniques