Trajectorial Martingales and Worst Case Approach to Market Models

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Financial markets are an essential part of modern societies; they provide the needed capital to make a myriad of economical activities possible. Regrettably, the negative side of financial speculation as well as the availability of cheap and unlimited credit create undesirable side effects (e.g. market crashes, uncontrollable debt, etc). In short, there is a definite need for a better understanding of financial markets. ******Stochastic modeling in financial mathematics is presently being generalized to incorporate higher levels of uncertainty. This initiative reflects the current inability of models to incorporate all variables that affect market conditions as well as a reliable joint probability distribution. Along this modern line of research, we propose an approach that weakens dramatically basic stochastic modeling assumptions. Conclusions in such an approach are more robust as they are less dependent on prior modeling assumptions. Our work will concentrate on a set of trajectories that replace the path space of the stochastic process. A thorough investigation will be pursued of several fundamental mathematical constructions that are available in this setting without any prior probabilistic assumptions.******A trajectorial version of the fundamental notion of financial arbitrage offers the possibility to define a general notion of trajectorial martingale. The latter concept is the analogue of martingale processes and as such is poised to play a central role in our approach. Among many developments, we will construct a pathwise version of conditional expectations and will study the possibility to develop a trajectorial analogue of the different variants of pathwise stochastic integrals.******A number of fundamental analytical developments are possible in the proposed framework. In particular, there is a natural integration operator defined by superhedging, a financial based approximation that provides coverage under each eventuality (trajectory wise) and which is not associated to a classical (Kolmogorov-type) probability measure.******Our approach pays attention to individual trajectories and, as such, could also be labelled worst case. The latter concept gains relevance and specificity in each particular application where assuming an apriori probability distribution is unwarranted. Our proposal does not make any assumptions on probability distributions (i.e. measure) but explores basic results that can be obtained prior to introducing a measure. This allows to gain a conceptual understanding of the financial meaning of new and established mathematical results that are obtained in our framework as they are interpreted without the language of probabilities. We will explore the financial implications of our new conceptual approach and will propose several market model constructions.