A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm

TL;DR

This paper introduces a reduced-basis variance reduction technique for efficiently computing parametrized expectations of stochastic differential equations, significantly improving computational speed in practical applications.

Contribution

It proposes two algorithms that pre-compute control variates offline using a greedy method, enabling fast online variance reduction for many parameter values.

Findings

Effective variance reduction demonstrated in option pricing calibration.

Significant computational savings in Langevin equation simulations.

Method applicable to various parametrized stochastic models.

Abstract

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.