Preintegration via Active Subspace

Abstract

Preintegration is an extension of conditional Monte Carlo to quasi–Monte Carlo and randomized quasi–Monte Carlo. Conditioning can reduce but not increase the variance in Monte Carlo. For quasi–Monte Carlo it can bring about improved regularity of the integrand with potentially greatly improved accuracy. We show theoretically that, just as in Monte Carlo, preintegration can reduce but not increase the variance when one uses scrambled net integration. Preintegration is ordinarily done by integrating out one of the input variables to a function. In the common case of a Gaussian integral one can also preintegrate over any linear combination of variables. For continuous functions that are differentiable almost everywhere, we propose to choose the linear combination by the first principal component in an active subspace decomposition. We show that the lead eigenvector in an active subspace decomposition is closely related to the vector that maximizes a computationally intractable criterion using a Sobol’ index. A numerical example of Asian option pricing finds that this active subspace preintegration strategy is competitive with preintegrating the first principal component of the Brownian motion, which is known to be very effective. The new method outperforms others on some basket and rainbow options where there is no generally accepted counterpart to the principal components construction.