We study the problem of reducing the variance of Monte Carlo estimators
through performing suitable changes of the sampling measure which are induced
by feedforward neural networks. To this end, building on the concept of vector
stochastic integration, we characterize the Cameron-Martin spaces of a large
class of Gaussian measures which are induced by vector-valued continuous local
martingales with deterministic covariation. We prove that feedforward neural
networks enjoy, up to an isometry, the universal approximation property in
these topological spaces. We then prove that sampling measures which are
generated by feedforward neural networks can approximate the optimal sampling
measure arbitrarily well. We conclude with a comprehensive numerical study
pricing path-dependent European options for asset price models that incorporate
factors such as changing business activity, knock-out barriers, dynamic
correlations, and high-dimensional baskets