Variance Reduction and Stochastic Differential Equations.

Summary This chapter deepens the variance reduction subject, and focuses on the Monte Carlo methods for deterministic parabolic partial differential equations. This topic requires advanced notions in stochastic calculus, particularly the Girsanov theorem, which we state and discuss first. We strongly emphasize that universal techniques do not exist: most often, effective variance reduction methods depend on the numerical analyst’s knowledge and experience. We will see that it is rather easy to construct perfect variance reduction methods which are irrelevant from a numerical point of view. a contrario, the construction of an effective method often lies on the approximation of a perfect method, the approximation method needing to be adapted to each particular case. Interesting examples can be found in Duffie and Glynn (Ann. Appl. Probab. 5(4), 897–905, 1995).