UVic Dynamical Systems Seminar Series: Sebastian Ferrando

We provide a brief motivational overview of recent developments of extensions of stochastics tools to deal with uncertainty. These are: Peng's nonlinear expectations and Ito's calculus without probabilities. We then describe a non-probabilistic version of a supermartingale theory closely motivated by financial considerations of no-arbitrage. The basic object replacing the classical filtered probability space is a structured trajectory set which allows the definition of conditional outer integrals as well as null sets. The conditional outer integrals are non linear functionals that allow to circumvent the linearity of the classical conditional expectations in proofs and definitions. Integrability notions emerge in our setting through non-classical conditional integral operators that lead to the special case of non-probabilistic martingales. One can define non-probabilistic supermartingales and prove analogous of classical results like: Doob's optional sampling theorem, Dubin's upcrossing inequalities and Doob's a.e. convergence for non-negative supermartingales. All constructions and results have a hedging and superhedging interpretation and there is a direct way in which the new results generalize the classical case. Null sets appearing in the results have a financial interpretation and are handled in a more concrete way than in the classical theory.