In this talk, we introduce deep learning schemes for forward–backward systems whose backward component is a backward stochastic Volterra integral equation (BSVIE). BSVIEs generalize classical BSDEs by introducing a second time variable, and arise naturally in recursive utilities with memory, time-inconsistent stochastic control, and dynamic risk measures.
Building on deep solvers for BSDEs, we develop backward learning schemes adapted to the two-time structure of BSVIEs. When the forward process is Markovian, the solution is given by deterministic functions of the two time variables and the forward state, which we approximate with neural networks. We prove the algorithm's convergence by decomposing the error into a time-discretization error and a neural network approximation error, and validate it numerically.
When the forward process is itself of Volterra type, the Markovian structure is lost, and the solution depends on the entire past trajectory. Using a path-dependent PDE representation, we approximate the solution with neural networks taking path signatures as inputs, and test the method on examples with explicit solutions.
Based on joint works with Nacira Agram, and with Nacira Agram and Mounir Zebbar.