  We present a novel computational approach for quadratic hedging in a
high-dimensional incomplete market. This covers both mean-variance hedging and
local risk minimization. In the first case, the solution is linked to a system
of BSDEs, one of which being a backward stochastic Riccati equation (BSRE); in
the second case, the solution is related to the F\"olmer-Schweizer
decomposition and is also linked to a BSDE. We apply (recursively) a deep
neural network-based BSDE solver. Thanks to this approach, we solve
high-dimensional quadratic hedging problems, providing the entire hedging
strategies paths, which, in alternative, would require to solve high
dimensional PDEs. We test our approach with a classical Heston model and with a
multi-dimensional generalization of it.
