Title:

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

Abstract:

We develop a reduced-basis approach for the efficient computation

of a large number of expected values using the control variate method to reduce the variance.

Two algorithms are proposed to compute online, through a cheap reduced-basis approximation,

the numerous (parametrized) control variates

for a large number of expectations of a functional of a parametrized It\^o stochastic process

(solution to a parametrized stochastic differential equation).

For each algorithm, a reduced basis is pre-computed offline,

following a Greedy procedure, which minimizes the variance among a trial sample of expectations.

Numerical results in situations relevant to practical applications

(the calibration of volatility in option pricing, and

the velocity-gradient-driven evolution of a vector field of FENE dumbbells)

illustrate the efficiency of the method.