Pathwise CVA regressions with oversimulated defaults

We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes (X,Y)$(X,Y)$. Here an exogenous component Y (Markov by itself) is time‐consuming to simulate, while the endogenous component X (jointly Markov with Y) is quick to simulate given Y, but is responsible for most of the variance of the simulated payoff. To address the related variance issue, we introduce a conditionally independent, hierarchical simulation scheme, where several paths of X are simulated for each simulated path of Y. We analyze the statistical convergence of the regression learning scheme based on such block‐dependent data. We derive heuristics on the number of paths of Y and, for each of them, of X, that should be simulated. The resulting algorithm is implemented on a graphics processing unit (GPU) combining Python/CUDA and learning with PyTorch. A CVA case study with a nested Monte Carlo benchmark shows that the hierarchical simulation technique is key to the success of the learning approach.