The a posteriori error bounds developed in Deep xVA Solver - A Neural Network Based Counterparty Credit Risk Management Framework provide computable estimates of the approximation error for neural network BSDE solvers, building on earlier results by Han and Long (2020), Bender and Steiner (2013), and Reisinger et al. (2020). These bounds are “a posteriori” in the sense that they depend on quantities observable after training — specifically, the terminal loss of the neural network optimisation — rather than on unknown network approximation capacities.
For the Deep BSDE Solver applied to a decoupled FBSDE, the fundamental estimate (Eq. 3.10) states that sup E|Y_t - Y_t^approx|^2 + integral of E|Z_t - Z_t^approx|^2 dt is bounded by C * (Delta_t + terminal_loss), where C depends only on model parameters (not on the time step or network architecture). This means that if the terminal loss is small (i.e., the network successfully minimises E[|g(X_T) - Y_T^approx|^2]), then both the value and control approximations are accurate uniformly in time. A complementary bound (Eq. 3.11) shows that the Y-error alone is controlled by the initial value error plus the integrated Z-error, which is practically useful since the initial value xi typically converges more easily.
For the non-recursive xVA setting, the paper derives (Eq. 3.15) that the error in the adjustment integral is bounded by C * (sqrt(Delta_t) + sum of sqrt(terminal_losses_m)), combining the quadrature error from the 1/2-Holder continuity of the integrand with the BSDE approximation errors from each portfolio constituent. For the recursive case (Algorithm 3), the overall xVA error depends additionally on the terminal loss of the xVA BSDE itself, propagating errors from both stages of the two-step procedure.
Key Details
- The constant C in the bounds depends on Lipschitz constants of the BSDE coefficients, the time horizon T, and the number of portfolio constituents M, but not on the dimension d or the network size
- The terminal loss E[|g(X_T) - Y_T^approx|^2] is directly reported by the training algorithm, making the bounds practically computable
- For the call option example, the initial value error (in xi) is negligible compared to the control error (in rho), consistent with the bound structure
- The quadrature error for non-recursive adjustments is sqrt(Delta_t), not Delta_t, due to the limited regularity (1/2-Holder) of the exposure integrand
- Extension to stochastic rates requires care with non-Lipschitz terms; a transformation (Eq. B.2) can eliminate the drift term rt * Vt in the clean value BSDE
- For fully coupled (recursive) BSDEs, the analysis extends via results of Reisinger et al. (2020) for McKean-Vlasov FBSDEs