A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm
For practitioners needing fast computation of many parametrized expectations (e.g., in finance or physics), this method offers a variance reduction technique that is computationally cheaper than traditional Monte Carlo.
This work develops a reduced-basis approach to efficiently compute parametrized expected values for stochastic differential equations, using control variates to reduce variance. Numerical results show efficiency in applications like option pricing and Langevin dynamics.
In this work, we develop a reduced-basis approach for the efficient 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 Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, 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.