New York Journal of Mathematics
Volume 24 (2018), 702-738


S.E. Ferrando and A.L. Gonzalez

Trajectorial martingale transforms. Convergence and integration

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Published: September 13, 2018
Keywords: Trajectorial martingales, worst case uncertainty, superhedging, non-lattice integration
Subject: Primary: 60G42, 60G48; Secondary: 60G17

Starting with a trajectory space, providing a non-stochastic analogue of a discrete time martingale process, we use the notion of super-replication to introduce definitions for null and full sets and the associated notion of a property holding almost everywhere (a.e.). The latter providing what can be seen as the worst case analogue of sets of measure zero in a stochastic setting. The a.e. notion is used to prove the pointwise convergence, on a full set of the original trajectory space, of the limit of a trajectorial transform sequence. The setting also allows to construct a natural integration operator which we study in detail.


The authors would like to thank Dr. Pedro Catuogno for stimulating conversations on the subject of the paper. S. Ferrando would like to thank Professor König for providing a personal copy of reference [7].

Author information

S.E. Ferrando:
Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, Ontario M5B 2K3, Canada.

A.L. Gonzalez:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, Mar del Plata 7600, Argentina.