DSAS Colloquium Talk - Sebastian Ferrando

Title: Trajectorial martingales. Convergence and Integration.

Abstract: Starting with the motivation of the uncertainty present in financial mathematics models,
we introduce  trajectory spaces providing a non-stochastic analogue of a discrete time martingale process. We use the notion of super-replication to define null and full functions 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 with some detail.
The latter concept leads to conditional trajectorial integrals and a general notion of trajectorial martingale.