Conditional NonLattice Integration, Pricing and Superhedging
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 byproduct 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 nonlattice integrals. One could then study analogues of classical probabilistic notions in such nonclassical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.12072
 arXiv:
 arXiv:2105.12072
 Bibcode:
 2021arXiv210512072B
 Keywords:

 Mathematics  Probability;
 Quantitative Finance  Mathematical Finance
 EPrint:
 44 pages