The weak functional representation of historical martingales. (English) Zbl 1515.60149

Summary: 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 [S. N. Evans and E. A. Perkins, Ann. Probab. 23, No. 4, 1772–1815 (1995; Zbl 0852.60062)].


60G57 Random measures
60H05 Stochastic integrals
60J68 Superprocesses


Zbl 0852.60062
Full Text: DOI


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