65 research outputs found
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 -H\"older continuous for all . 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
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