2015/08/26 by Anis Al Gerbi, Benjamin Jourdain, Gerbi, Anis Al +3
Economics, Econometrics and Finance · Mathematics · #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Probability (math.PR) #math.PR #q-fin.CP
paper · pdf · doi:10.48550/arxiv.1508.06492
arxiv created 2015/10/07 · arxiv updated 2015/10/08
In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity O(ε-2) for the precision ε. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order 1 to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.