Derivative-Informed Dimension Reduction of Stochastic Differential Equations: A Path-Space Sensitivity Approach

Abstract

Uncertainty in the drift of a stochastic differential equation (SDE) induces uncertainty in statistical observables defined as expected functionals on path space. We propose a derivative-informed method for identifying low-dimensional structure in the drift-to-observable map, with the aim of improving the tractability of uncertainty quantification. This approach can be viewed as a generalization of the active subspace method to infinite-dimensional input spaces and to path-space observables of SDEs. Representing uncertainty in the drift by random perturbations in a Hilbert space, we construct a path-space sensitivity operator using first-order Fréchet derivatives of the observable with respect to the perturbation. We establish conditions under which the leading eigenfunctions of this operator correspond to directions of maximum mean squared sensitivity of the observable, thus enabling low-dimensional approximation with certified error bounds. We further introduce a martingale control variate of the Fréchet derivative that reduces error in sample-based estimators of the operator. Using a Galerkin scheme for numerical discretization, we show that reduced representations of the drift from the derivative-informed basis yield more accurate approximations of mean first hitting times of SDEs than the Karhunen-Loève expansion of the same truncation order.