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

In this work, we develop a reduced-basis approach for the e cient 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 Itô stochastic process (solution to a parametrized stochastic di erential equation). For each algorithm, a reduced basis of control variates is pre-computed ofine, 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 eld following a Langevin equation from kinetic theory) illustrate the e ciency of the method.