18,214 research outputs found
On convexity of solutions of ordinary differential equations
We prove a result on the convex dependence of solutions of ordinary
differential equations on an ordered finite-dimensional real vector space with
respect to the initial data.Comment: 10 page
Optimal Transport and Skorokhod Embedding
The Skorokhod embedding problem is to represent a given probability as the
distribution of Brownian motion at a chosen stopping time. Over the last 50
years this has become one of the important classical problems in probability
theory and a number of authors have constructed solutions with particular
optimality properties. These constructions employ a variety of techniques
ranging from excursion theory to potential and PDE theory and have been used in
many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts
from optimal mass transport. In analogy to the celebrated article of Gangbo and
McCann on the geometry of optimal transport, we establish a geometric
characterization of Skorokhod embeddings with desired optimality properties.
This leads to a systematic method to construct optimal embeddings. It allows
us, for the first time, to derive all known optimal Skorokhod embeddings as
special cases of one unified construction and leads to a variety of new
embeddings. While previous constructions typically used particular properties
of Brownian motion, our approach applies to all sufficiently regular Markov
processes.Comment: Substantial revision to improve the readability of the pape
Model-independent pricing with insider information: a Skorokhod embedding approach
In this paper, we consider the pricing and hedging of a financial derivative
for an insider trader, in a model-independent setting. In particular, we
suppose that the insider wants to act in a way which is independent of any
modelling assumptions, but that she observes market information in the form of
the prices of vanilla call options on the asset. We also assume that both the
insider's information, which takes the form of a set of impossible paths, and
the payoff of the derivative are time-invariant. This setup allows us to adapt
recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and
a monotonicity principle, which enables us to determine geometric properties of
the optimal models. Moreover, we show that this setup is powerful, in that we
are able to find analytic and numerical solutions to certain pricing and
hedging problems
Quantum Feynman-Kac perturbations
We develop fully noncommutative Feynman-Kac formulae by employing quantum
stochastic processes. To this end we establish some theory for perturbing
quantum stochastic flows on von Neumann algebras by multiplier cocycles.
Multiplier cocycles are constructed via quantum stochastic differential
equations whose coefficients are driven by the flow. The resulting class of
cocycles is characterised under alternative assumptions of separability or
Markov regularity. Our results generalise those obtained using classical
Brownian motion on the one hand, and results for unitarily implemented flows on
the other.Comment: 27 pages. Minor corrections to version 2. To appear in the Journal of
the London Mathematical Societ
Rotational cooling of trapped polyatomic molecules
Controlling the internal degrees of freedom is a key challenge for
applications of cold and ultracold molecules. Here, we demonstrate
rotational-state cooling of trapped methyl fluoride molecules (CH3F) by
optically pumping the population of 16 M-sublevels in the rotational states
J=3,4,5, and 6 into a single level. By combining rotational-state cooling with
motional cooling, we increase the relative number of molecules in the state
J=4, K=3, M=4 from a few percent to over 70%, thereby generating a
translationally cold (~30mK) and nearly pure state ensemble of about 10^6
molecules. Our scheme is extendable to larger sets of initial states, other
final states and a variety of molecule species, thus paving the way for
internal-state control of ever larger molecules
Pathwise super-replication via Vovk's outer measure
Since Hobson's seminal paper [D. Hobson: Robust hedging of the lookback
option. In: Finance Stoch. (1998)] the connection between model-independent
pricing and the Skorokhod embedding problem has been a driving force in robust
finance. We establish a general pricing-hedging duality for financial
derivatives which are susceptible to the Skorokhod approach.
Using Vovk's approach to mathematical finance we derive a model-independent
super-replication theorem in continuous time, given information on finitely
many marginals. Our result covers a broad range of exotic derivatives,
including lookback options, discretely monitored Asian options, and options on
realized variance.Comment: 18 page
A wave function based ab initio non-equilibrium Green's function approach to charge transport
We present a novel ab initio non-equilibrium approach to calculate the
current across a molecular junction. The method rests on a wave function based
description of the central region of the junction combined with a tight binding
approximation for the electrodes in the frame of the Keldysh Green's function
formalism. In addition we present an extension so as to include effects of the
two-particle propagator. Our procedure is demonstrated for a dithiolbenzene
molecule between silver electrodes. The full current-voltage characteristic is
calculated. Specific conclusions for the contribution of correlation and
two-particle effects are derived. The latter are found to contribute about 5%
to the current. The order of magnitude of the current coincides with
experiments.Comment: 21 pages, 3 figure
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