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.

Published at http://dx.doi.org/10.3150/15-BEJ735 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
• Rough path recursions and diffusion approximations
• A Fourier approach to pathwise stochastic integration

Cited by in corpus (6)

• Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity
• Pathwise integration with respect to paths of finite quadratic variation
• Martingale optimal transport duality
• Rough path recursions and diffusion approximations
• Getting rich quick with the Axiom of Choice
• Gamma Hedging and Rough Paths