Stochastic integration and differential equations for typical paths

TL;DR

This paper develops a pathwise approach to defining stochastic integrals and solving stochastic differential equations in infinite-dimensional Hilbert spaces without relying on probability, using outer measures and path properties.

Contribution

It introduces a novel outer measure framework that enables direct construction of stochastic integrals and solutions to SDEs for typical paths in infinite-dimensional spaces.

Findings

Constructed continuous stochastic integrals for typical paths.

Enabled solving SDEs in a model-free, pathwise manner.

Provided two methods for pathwise stochastic calculus in infinite dimensions.

Abstract

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega$ but also in $\int\omega\,d\omega:=\omega^2 -\langle \omega \rangle$ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega^2 -\langle \omega \rangle$ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.