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We study the problem of reducing the variance of Monte Carlo estimatorsthrough performing suitable changes of the sampling measure which are inducedby feedforward neural networks. To this end, building on the concept of vectorstochastic integration, we characterize the Cameron-Martin spaces of a largeclass of Gaussian measures which are induced by vector-valued continuous localmartingales with deterministic covariation. We prove that feedforward neuralnetworks enjoy, up to an isometry, the universal approximation property inthese topological spaces. We then prove that sampling measures which aregenerated by feedforward neural networks can approximate the optimal samplingmeasure arbitrarily well. We conclude with a comprehensive numerical studypricing path-dependent European options for asset price models that incorporatefactors such as changing business activity, knock-out barriers, dynamiccorrelations, and high-dimensional baskets.

Importance sampling for option pricing with feedforward neural networks