Open Access
2022 The weak functional representation of historical martingales
Christian Mandler, Ludger Overbeck
Author Affiliations +
Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP492


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 [7].

Funding Statement

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.


Download Citation

Christian Mandler. Ludger Overbeck. "The weak functional representation of historical martingales." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 19 August 2021; Accepted: 9 October 2022; Published: 2022
First available in Project Euclid: 19 October 2022

MathSciNet: MR4498571
zbMATH: 1515.60149
Digital Object Identifier: 10.1214/22-ECP492

Primary: 60G57 , 60H05 , 60J68

Keywords: historical Brownian motion , Martingale representation , Superprocesses

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