Winsome Huei-Wen Teng from National Chiao Tung University.
Paper: On a Novel Spherical Monte Carlo Method via Group Representation
Accurate and efficient calculation of d-dimensional integrals for large d is of crucial importance in various scientific disciplines. Via spherical transformation, standard spherical Monte Carlo estimators consist of independent radii and a set of unit vectors uniformly distributed on a unit sphere. A random orthogonal group is used to rotate a set of unit vectors simultaneously, and can be generated by applying the Gram–Schmidt procedure to a d × d matrix with i.i.d. standard normal random variables as entries. The generation of a random orthogonal group is however computationally demanding. To overcome this problem, this paper proposes a novel spherical Monte Carlo approach via group representation: By constructing a subgroup of the orthogonal groups, the spherical integral is calculated using the group orbit of a random unit vector. In this case, the generation of a random unit vector only needs d i.i.d. standard normal random variable. The proposed method outperforms existing methods in terms of computation efficiency in high-dimensional cases. Theoretical properties of the proposed subset are provided. Extensive numerical experiments with applications in finance confirm our claims.