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