## Bernoulli

• Bernoulli
• Volume 22, Number 4 (2016), 2486-2520.

### Pathwise stochastic integrals for model free finance

#### 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.

#### Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2486-2520.

Dates
Revised: March 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297687

Digital Object Identifier
doi:10.3150/15-BEJ735

Mathematical Reviews number (MathSciNet)
MR3498035

Zentralblatt MATH identifier
1346.60078

#### Citation

Perkowski, Nicolas; Prömel, David J. Pathwise stochastic integrals for model free finance. Bernoulli 22 (2016), no. 4, 2486--2520. doi:10.3150/15-BEJ735. https://projecteuclid.org/euclid.bj/1462297687

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