By iterating a martingale representation result a homogeneous chaos expansion is obtained. Using the martingale representation, the integration-by-parts formula of the Malliavin calculus is derived using properties of stochastic flows. The infinite-dimensional calculus of variations is not required.
Robert J. Elliott. Michael Kohlmann. "Integration by Parts, Homogeneous Chaos Expansions and Smooth Densities." Ann. Probab. 17 (1) 194 - 207, January, 1989. https://doi.org/10.1214/aop/1176991504