  Motivated by the equations of cross valuation adjustments (XVAs) in the
realistic case where capital is deemed fungible as a source of funding for
variation margin, we introduce a simulation/regression scheme for a class of
anticipated BSDEs, where the coefficient entails a conditional expected
shortfall of the martingale part of the solution. The scheme is explicit in
time and uses neural network least-squares and quantile regressions for the
embedded conditional expectations and expected shortfall computations. An a
posteriori Monte Carlo validation procedure allows assessing the regression
error of the scheme at each time step. The superiority of this scheme with
respect to Picard iterations is illustrated in a high-dimensional and hybrid
market/default risks XVA use-case.
