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# Communications in Mathematical Sciences

## Volume 8 (2010)

### Number 3

### Special Issue: Mathematical Issues of Complex Fluids

### A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm

Pages: 735 – 762

DOI: http://dx.doi.org/10.4310/CMS.2010.v8.n3.a7

#### Authors

#### Abstract

In this work, we develop a reduced-basis approach for the deficient 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 differential equation). For each algorithm, a reduced basis of control variates is pre-computed offine, 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.

#### Keywords

Variance reduction; stochastic differential equations; reduced-basis methods

#### 2010 Mathematics Subject Classification

60H10, 65C05