Two extensions of the Ito integral are developed, and put in the perspective of derivative operators in the Malliavin calculus. The divergence operator, $\delta$, is constructed, and its properties and action on these two extended integrals are described. Discussion of iterated stochastic integrals and the extended stochastic integrals as functions of their upper limits is also included.
Marc A. Berger. "A Malliavin-Type Anticipative Stochastic Calculus." Ann. Probab. 16 (1) 231 - 245, January, 1988. https://doi.org/10.1214/aop/1176991897