Abstracts

The introduction of Knightian uncertainty in mathematical models for finance has recently renew the attention on foundational issues such as the existence of option pricing rules or super-hedging theorems. In this context an agent allows for the possibility of describing a certain market with a probabilistic model but he can’t be sure about a single reference probability, hence, a whole class of reference probabilities is taken into account. Although the literature on this topic is rapidly growing it is not clear yet what is a good notion of arbitrage opportunity and the consequent properties on martingale measures. In the present work we slightly change the point of view avoiding the necessity of fixing a subset of probability measures, which might be problematic in some settings. Given a certain market model, described by a discrete time stochastic process, we study the intrinsic properties of the model, independently on any reference probability. In particular two questions arises naturally: 1) which markets are feasible, in the sense that the properties of the market are nice for most probabilistic models? 2) which are the markets that exhibits no arbitrage for most probabilistic models?
A good notion for “most” probabilistic models is clearly needed and it is undertaken in this work from a topological point of view. Different results are obtained depending on the coarseness of the topology under consideration.
A key property is the existence of martingale measures. An important difference from the classical case is that we are not providing equivalent conditions in terms of absence of arbitrage opportunities, we are studying, instead, the structural properties of the market needed for the existence of such measures.
Once this minimal property of the market is guaranteed, we introduce definitions of arbitrage as trading strategies that work for most probabilistic models, in the spirit of the rest of the work. Additional properties on the martingale measures are retrieved under the No arbitrage hypothesis.