  This paper develops a new efficient scheme for approximations of expectations
of the solutions to stochastic differential equations (SDEs). In particular, we
present a method for connecting approximate operators based on an asymptotic
expansion with multidimensional Malliavin weights to compute a target
expectation value precisely. The mathematical validity is given based on
Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical
experiments for option pricing under local and stochastic volatility models
confirm the effectiveness of our scheme. Especially, our weak approximation
substantially improves the accuracy at deep Out-of-The-Moneys (OTMs).
