Junichi Imai: Integrated quasi-Monte Carlo methods with dimension reduction and discontinuity realignment

The dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi-Monte Carlo (QMC) method.
The first factor implies that the QMC method can have high performance on
problem with low effective dimension even though its nominal
dimension can be very large.
The second factor suggests that the QMC method can be very effective on problem that is smooth or even on problem that is discontinuous as long as its discontinuities are parallel to the axes.
Motivated by these findings, methods such as the linear transformation (LT) method of Imai and Tan (2006) and the orthogonal transformation (OT) method of Wang and Tan (2013) have been proposed to increase the efficiency of QMC. However, both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the LT method enhances QMC by effective dimension reduction while the OT method accomplishes the same objective by discontinuity realignment. On the other hand there are many problems in finance that are of both high dimensionality and discontinuous. The main objective of this paper is to propose an efficient QMC method for handling problems of this kind. We first prove that the OT method is a special case of the LT method. We then show that by integrating the LT and OT methods, the proposed method has the advantage of addressing both dimensionality and discontinuity concurrently.  The numerical examples indicate that relative to the
  standard QMC, the proposed hybrid method is extremely effective and could attain a variance reduction as high as several thousands times.
We further discuss a possible extension of the proposed method to the case under a Le’vy process and demonstrate its competitive efficiency using some additional numerical examples.