Measure-Valued Martingales, Martingale Optimal Transport & Model-Independent Finance

I propose to study three interrelated problems at the intersection of stochastic analysis, probability theory and stochastic control theory.

Specifically, the aim is to develop a suitable stochastic calculus and control theory for measure-valued martingales – a class of non-vector valued and infinite-dimensional stochastic processes; to define suitable notions of distances between stochastic processes when taking the flow of information into appropriate account; and to study quasi-convex risk measures with particular emphasis on their dynamic consistency.The problems are unified by their relevance for addressing an open problem in mathematical finance – namely for understanding sensitivity contra stability with respect to market modelling.

Classical approaches to the subject take their outset in the specification of a stochastic market model; while we can address most financial questions for very general models, a proper understanding for how the derived answers depend on the model itself is still lacking.

It is critical for an accurate evaluation of financial guidance and for controlling our exposure to model misspecification.The proposed problems are also of key mathematical interest in their own right and greatly extend existing theory.

Further applications include Martingale Optimal Transport, Skorokhod Embedding Problems, Functional Inequalities and Filtration Enlargement.The project will be carried out by me and a PhD student which I intend to hire on the grant.