Pathwise stochastic integration and applications to the theory of continuous trading

Abstract

We develop a pathwise construction of stochastic integrals relative to continuous martingales. The key to the construction is an almost sure approximation technique which associates a sequence of finitely generated filtrations (“skeleton filtrations”) and a sequence of simple stochastic processes (“skeleton processes”) to a given continuous martingale and its underlying filtration (which is also assumed to be “continuous”). The pathwise stochastic integral can then be defined along such a “skeleton approximation” and almost-sure convergence follows from a certain completeness property of the skeleton approximation. The limit is the pathwise stochastic integral and it agrees with the integral obtained through the usual Itô approach. The pathwise stochastic integration theory is applied to the analysis of stochastic models of security markets with continuous trading. A convergence theory for continuous market models with exogenously given equilibrium prices is obtained which enables one to view an idealized economy (i.e. a continuous market model) as an almost sure limit of “real-life” economies (i.e. finite market models). Since finite market models are well understood, this convergence sheds light on features of the continuous market model, such as completeness.