A weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in .
The first author would like to thank the Deutsche Forschungsgemeinschaft for its financial support.
The authors would like to thank the referees for their insightful comments.
"The weak functional representation of historical martingales." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP492