[Paper Review] Pathwise stochastic integrals for model free finance

Nicolas Perkowski, David J. Pr omel|RePEc: Research Papers in Economics|Nov 24, 2013

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces two model-free stochastic integration frameworks in continuous-time financial mathematics without probabilistic assumptions. The first uses a topology induced by Vovk's outer measure (minimal superhedging price), enabling a continuous extension of the Itô integral to càdlàg adapted processes. The second leverages controlled rough path theory, proving that typical price paths admit an Itô rough path and that the rough path integral arises as a limit of non-anticipating Riemann sums—offering a Banach space structure while preserving financial intuition.

ABSTRACT

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every "typical price path" has a naturally associated Itô rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory. Both approaches are based entirely on financial arguments and do not require any probabilistic structure.

Motivation & Objective

• To develop a rigorous stochastic integration theory in frictionless, model-free financial mathematics without assuming probabilistic or semimartingale structures.
• To overcome the limitations of prior approaches—such as bounded variation or quadratic variation restrictions—by constructing a more general and continuous integration framework.
• To reconcile the Banach space structure of rough path theory with the financial interpretability of non-anticipating trading strategies.
• To justify the use of controlled rough path integrals in finance by proving they emerge as limits of non-anticipating Riemann sums, thus restoring their financial plausibility.

Proposed method

• Uses Vovk’s game-theoretic outer measure, defined as the pathwise minimal superhedging price, to induce a topology on the space of continuous paths.
• Defines a model-free Itô integral as the continuous extension of the step function integral under this topology, applicable to càdlàg adapted processes.
• Establishes that every typical price path (in Vovk’s sense) naturally carries an Itô rough path, enabling the use of controlled rough path theory.
• Proves that the controlled rough path integral is the limit of non-anticipating Riemann sums under weak regularity conditions, avoiding compensated sums.
• Applies Davie’s criterion to show convergence of standard Riemann sums to the rough path integral under Hölder and area regularity conditions.
• Demonstrates that the rough path integral extends classical frameworks: bounded variation, Young, and Föllmer’s pathwise Itô integrals.

Experimental results

Research questions

• RQ1 Can a continuous, model-free stochastic integral be constructed without assuming probabilistic or semimartingale structures on price paths?
• RQ2 Does the controlled rough path integral, typically defined via compensated Riemann sums, admit a financial interpretation as a limit of non-anticipating Riemann sums?
• RQ3 Can the model-free Itô integral be extended to càdlàg adapted integrands while preserving continuity under a financial topology?
• RQ4 Is Föllmer’s pathwise Itô integral a special case of the controlled rough path integral in the model-free setting?
• RQ5 Can the rough path integral be rigorously justified in finance by showing convergence of non-anticipating Riemann sums without compensation?

Key findings

• The model-free Itô integral is continuous with respect to a topology induced by Vovk’s outer measure, enabling extension to càdlàg adapted integrands.
• Every typical price path in Vovk’s sense admits a natural Itô rough path, enabling the application of controlled rough path theory in model-free finance.
• The controlled rough path integral is shown to be the limit of non-anticipating Riemann sums under weak regularity conditions, resolving a key philosophical objection to its use.
• The rough path integral generalizes and unifies prior methods: bounded variation, Young, and Föllmer’s pathwise Itô integrals.
• Davie’s criterion implies that the rough path integral converges as a standard Riemann sum under Hölder and area regularity conditions, validating its use without compensation.
• The paper provides the first rigorous proof that Föllmer’s pathwise Itô integral is a special case of the controlled rough path integral in the model-free context.