Conditional non-lattice integration, pricing, and superhedging

Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated to a measure.The base space, denoted by S and called a trajectory space, substitutes the set Ω in probability theory and provides a fundamental structure via conditional subsets S (S,j) that allows the definition of conditional integrals.The set S naturally embodies a weak no-arbitrage hypothesis which in turn is key to establishing the basic properties needed to develop the theory of integration.The constructed conditional integrals can be interpreted as the required investment, at each conditioning node, to hedge an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used.The a.e.notion is defined by means of financial considerations and it does not rely on measure theory.The integral is not classical due to the fact that the elementary vector space of portfolio payoffs is not a vector lattice.In contrast to a classical stochastic setting, where price processes are associated to conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals.In particular, no measurability assumptions are needed.Besides the central role of S, key ingredients in our approach are conditional superhedging operators that act as conditional outer integrals, the associated superhedging norms provide countable subadditivity and allow to define null events.