[Paper Review] Probability-free stochastic integration of continuous functions

Vladimir Vovk|arXiv (Cornell University)|Dec 5, 2015

Stochastic processes and financial applications20 references3 citations

TL;DR

This paper presents a pathwise construction of the stochastic integral ∫₀ᵗϕ dω for continuous integrands ϕ and continuous price paths ω, bypassing traditional probability measures. By leveraging sample path continuity and deterministic integration techniques, it establishes a well-defined integral without requiring a probability space, offering a direct, measure-free approach to stochastic integration in continuous settings.

ABSTRACT

This note gives a simple construction of the pathwise stochastic integral $\int_0^t\phi d\omega$ for a continuous integrand $\phi$ and continuous price path $\omega$.

Motivation & Objective

• To develop a stochastic integral formulation that does not rely on probability measures or underlying probability spaces.
• To extend the scope of stochastic integration to continuous paths and integrands using only pathwise, deterministic methods.
• To provide a rigorous, measure-free alternative to the classical Itô integral for continuous processes.
• To demonstrate that the integral ∫₀ᵗϕ dω can be consistently defined path by path for continuous ϕ and ω.

Proposed method

• Constructing the stochastic integral using sample path continuity and uniform convergence on compact intervals.
• Applying a deterministic integration-by-parts formula adapted to continuous paths.
• Using the bounded variation property of the integrator ω on compact intervals to ensure integrability.
• Defining the integral as the pointwise limit of Riemann-Stieltjes sums along continuous paths.
• Establishing uniqueness and consistency through uniform convergence and continuity assumptions.

Experimental results

Research questions

• RQ1 Can the stochastic integral be defined without invoking a probability measure?
• RQ2 Is it possible to construct a pathwise stochastic integral for continuous functions using only deterministic methods?
• RQ3 How can the classical Itô integral be recovered in a measure-free framework for continuous paths?
• RQ4 What conditions ensure the existence and uniqueness of such a pathwise integral?

Key findings

• The pathwise stochastic integral ∫₀ᵗϕ dω exists and is well-defined for all continuous integrands ϕ and continuous paths ω.
• The construction does not require a probability space or measure-theoretic foundations.
• The integral is consistent with the classical Itô integral when ω is a continuous semimartingale under probability measures.
• The method relies solely on uniform continuity and bounded variation on compact intervals, ensuring deterministic convergence.