  The present article revisits the Diffusion Operator Integral (DOI) variance
reduction technique originally proposed in Heath and Platen (2002) and extends
its theoretical concept to the pricing of American-style options under
(time-homogeneous) L\'evy stochastic differential equations. The resulting Jump
Diffusion Operator Integral (JDOI) method can be combined with numerous Monte
Carlo based stopping-time algorithms, including the ubiquitous least-squares
Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996),
Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical
derivations under a concrete, though very general jump-diffusion stochastic
volatility dynamics and test the resulting LSMC based version of the JDOI
method. The results provide evidence of a strong variance reduction when
compared with a simple application of the LSMC algorithm and proves that
applying our technique on top of Monte Carlo based pricing schemes provides a
powerful way to speed-up these methods.
