2013/11/30 by Nicolas Perkowski, David J. Prömel · 1 citation
Mathematics · Economics, Econometrics and Finance · #math.PR #q-fin.GN
paper · pdf · doi:10.3150/15-bej735
published as Bernoulli 2016, Vol. 22, No. 4, 2486-2520 · 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)
arxiv created 2016/06/27 · arxiv updated 2016/06/28
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.