Conditional Non-Lattice Integration, Pricing and Superhedging

TL;DR

This paper introduces a novel integration framework inspired by financial markets that does not rely on measures, allowing for the modeling of prices and hedging strategies in a non-lattice setting.

Contribution

It develops a non-measure-based integration theory on trajectory spaces, enabling the analysis of prices and hedging without classical stochastic assumptions.

Findings

Conditional integrals represent required investments for hedging.

The theory applies to non-lattice payoff spaces, diverging from classical models.

Prices are given by conditional non-lattice integrals, not expectations.

Abstract

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 $\mathcal{S}$ 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 to 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 non-classical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.