329 research outputs found
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
Rough differential equations driven by signals in Besov spaces
Rough differential equations are solved for signals in general Besov spaces
unifying in particular the known results in H\"older and p-variation topology.
To this end the paracontrolled distribution approach, which has been introduced
by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular
PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is
extended from H\"older to Besov spaces. As an application we solve stochastic
differential equations driven by random functions in Besov spaces and Gaussian
processes in a pathwise sense.Comment: Former title: "Rough differential equations on Besov spaces", 37
page
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
Optimal extension to Sobolev rough paths
We show that every -valued Sobolev path with regularity
and integrability can be lifted to a Sobolev rough path in the
sense of T. Lyons provided . Moreover, we prove the existence of
unique rough path lifts which are optimal w.r.t. strictly convex functionals
among all possible rough path lifts given a Sobolev path. As examples, we
consider the rough path lift with minimal Sobolev norm and characterize the
Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a
suitable convex functional. Generalizations of the results to Besov spaces are
briefly discussed.Comment: Typos fixed. To appear in Potential Analysi
An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift
We solve the Skorokhod embedding problem for a class of Gaussian processes
including Brownian motion with non-linear drift. Our approach relies on solving
an associated strongly coupled system of Forward Backward Stochastic
Differential Equation (FBSDE), and investigating the regularity of the obtained
solution. For this purpose we extend the existence, uniqueness and regularity
theory of so called decoupling fields for Markovian FBSDE to a setting in which
the coefficients are only locally Lipschitz continuous
Duality for pathwise superhedging in continuous time
We provide a model-free pricing-hedging duality in continuous time. For a
frictionless market consisting of risky assets with continuous price
trajectories, we show that the purely analytic problem of finding the minimal
superhedging price of a path dependent European option has the same value as
the purely probabilistic problem of finding the supremum of the expectations of
the option over all martingale measures. The superhedging problem is formulated
with simple trading strategies, the claim is the limit inferior of continuous
functions, which allows for upper and lower semi-continuous claims, and
superhedging is required in the pathwise sense on a -compact sample
space of price trajectories. If the sample space is stable under stopping, the
probabilistic problem reduces to finding the supremum over all martingale
measures with compact support. As an application of the general results we
deduce dualities for Vovk's outer measure and semi-static superhedging with
finitely many securities
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
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