Conditioned Martingales

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.Comment: Corrected several typos, improved formulations. Accepted by Electronic Communications in Probability; Electronic Communications in Probability, 2012, Volume 17, Issue 4

Supermartingales as Radon-Nikodym densities and related measure extensions

Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.Comment: Published at http://dx.doi.org/10.1214/14-AOP956 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pathwise stochastic integrals for model free finance

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of It\^o'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 It\^o 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.Comment: 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

Local times for typical price paths and pathwise Tanaka formulas

Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $\alpha$-H\"older continuous for all $\alpha<1/2$. Additionally, we provide various generalizations of F\"ollmer's pathwise It\^o formula

Fractional Kolmogorov equations with singular paracontrolled terminal conditions

We consider backward fractional Kolmogorov equations with singular Besov drift of low regularity and singular terminal conditions. To treat drifts beyond the socalled Young regime, we assume an enhancement assumption on the drift and consider paracontrolled terminal conditions. Our work generalizes previous results on the equation from Cannizzaro, Chouk 2018 and Kremp, Perkowski 2022 to the case of singular paracontrolled terminal conditions and simultaneously treats singular and non-singular data in one concise solution theory. We introduce a paracontrolled solution space, that implies parabolic time and space regularity on the solution without introducing the socalled "modified paraproduct" from Gubinelli, Perkowski 2017. The tools developed in this article apply for general linear PDEs that can be tackled with the paracontrolled ansatz