PIMS - UVic Dynamics and Probability Seminar: Sebastian Ferrando

Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated with a measure. The base space, denoted by $\mathcal{S}$ and called a trajectory space, substitutes the set $\Omega$ in probability theory and provides a fundamental structure via conditional subsets $\mathcal{S}_{(S,j)}$ that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated with conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such a non-classical setting, we enumerate some of the possible results such as Doob's martingale convergence theorem and indicate an analogy with a dynamical system setting and the ergodic theorem.