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In this work, we develop a reduced-basis approach for the efficientcomputation of parametrized expected values, for a large number of parametervalues, using the control variate method to reduce the variance. Two algorithmsare proposed to compute online, through a cheap reduced-basis approximation,the control variates for the computation of a large number of expectations of afunctional of a parametrized Ito stochastic process (solution to a parametrizedstochastic differential equation). For each algorithm, a reduced basis ofcontrol variates is pre-computed offline, following a so-called greedyprocedure, which minimizes the variance among a trial sample of the outputparametrized expectations. Numerical results in situations relevant topractical applications (calibration of volatility in option pricing, andparameter-driven evolution of a vector field following a Langevin equation fromkinetic theory) illustrate the efficiency of the method.

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