647 research outputs found
Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models
Consider power utility maximization of terminal wealth in a 1-dimensional
continuous-time exponential Levy model with finite time horizon. We discretize
the model by restricting portfolio adjustments to an equidistant discrete time
grid. Under minimal assumptions we prove convergence of the optimal
discrete-time strategies to the continuous-time counterpart. In addition, we
provide and compare qualitative properties of the discrete-time and
continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research.
The final publication is available at springerlink.co
Fractional smoothness and applications in finance
This overview article concerns the notion of fractional smoothness of random
variables of the form , where is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages
On the harmonic measure of stable processes
Using three hypergeometric identities, we evaluate the harmonic measure of a
finite interval and of its complementary for a strictly stable real L{\'e}vy
process. This gives a simple and unified proof of several results in the
literature, old and recent. We also provide a full description of the
corresponding Green functions. As a by-product, we compute the hitting
probabilities of points and describe the non-negative harmonic functions for
the stable process killed outside a finite interval
From Relational Data to Graphs: Inferring Significant Links using Generalized Hypergeometric Ensembles
The inference of network topologies from relational data is an important
problem in data analysis. Exemplary applications include the reconstruction of
social ties from data on human interactions, the inference of gene
co-expression networks from DNA microarray data, or the learning of semantic
relationships based on co-occurrences of words in documents. Solving these
problems requires techniques to infer significant links in noisy relational
data. In this short paper, we propose a new statistical modeling framework to
address this challenge. It builds on generalized hypergeometric ensembles, a
class of generative stochastic models that give rise to analytically tractable
probability spaces of directed, multi-edge graphs. We show how this framework
can be used to assess the significance of links in noisy relational data. We
illustrate our method in two data sets capturing spatio-temporal proximity
relations between actors in a social system. The results show that our
analytical framework provides a new approach to infer significant links from
relational data, with interesting perspectives for the mining of data on social
systems.Comment: 10 pages, 8 figures, accepted at SocInfo201
Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law
We study the electronic transport properties of the Anderson model on a
strip, modeling a quasi one-dimensional disordered quantum wire. In the
literature, the standard description of such wires is via random matrix theory
(RMT). Our objective is to firmly relate this theory to a microscopic model. We
correct and extend previous work (arXiv:0912.1574) on the same topic. In
particular, we obtain through a physically motivated scaling limit an ensemble
of random matrices that is close to, but not identical to the standard transfer
matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson
symmetry classes \beta=1,2. In the \beta=2 class, the resulting conductance is
the same as the one from the ideal ensemble, i.e.\ from TUE. In the \beta=1
class, we find a deviation from TOE. It remains to be seen whether or not this
deviation vanishes in a thick-wire limit, which is the experimentally relevant
regime. For the ideal ensembles, we also prove Ohm's law for all symmetry
classes, making mathematically precise a moment expansion by Mello and Stone.
This proof bypasses the explicit but intricate solution methods that underlie
most previous results.Comment: Corrects and extends arXiv:0912.157
A stochastic network with mobile users in heavy traffic
We consider a stochastic network with mobile users in a heavy-traffic regime.
We derive the scaling limit of the multi-dimensional queue length process and
prove a form of spatial state space collapse. The proof exploits a recent
result by Lambert and Simatos which provides a general principle to establish
scaling limits of regenerative processes based on the convergence of their
excursions. We also prove weak convergence of the sequences of stationary joint
queue length distributions and stationary sojourn times.Comment: Final version accepted for publication in Queueing Systems, Theory
and Application
On truncated variation, upward truncated variation and downward truncated variation for diffusions
The truncated variation, , is a fairly new concept introduced in [5].
Roughly speaking, given a c\`adl\`ag function , its truncated variation is
"the total variation which does not pay attention to small changes of ,
below some threshold ". The very basic consequence of such approach is
that contrary to the total variation, is always finite. This is
appealing to the stochastic analysis where so-far large classes of processes,
like semimartingales or diffusions, could not be studied with the total
variation. Recently in [6], another characterization of was found.
Namely is the smallest total variation of a function which approximates
uniformly with accuracy . Due to these properties we envisage that
might be a useful concept to the theory of processes.
For this reason we determine some properties of for some well-known
processes. In course of our research we discover intimate connections with
already known concepts of the stochastic processes theory.
Firstly, for semimartingales we proved that is of order and
the normalized truncated variation converges almost surely to the quadratic
variation of the semimartingale as . Secondly, we studied the rate
of this convergence. As this task was much more demanding we narrowed to the
class of diffusions (with some mild additional assumptions). We obtained the
weak convergence to a so-called Ocone martingale. These results can be viewed
as some kind of large numbers theorem and the corresponding central limit
theorem.
All the results above were obtained in a functional setting, viz. we worked
with processes describing the growth of the truncated variation in time.
Moreover, in the same respect we also treated two closely related quantities -
the so-called upward truncated variation and downward truncated variation.Comment: Added Remark 6 and Remark 15. Some exposition improvement and fixed
constant
Relativistic diffusion processes and random walk models
The nonrelativistic standard model for a continuous, one-parameter diffusion
process in position space is the Wiener process. As well-known, the Gaussian
transition probability density function (PDF) of this process is in conflict
with special relativity, as it permits particles to propagate faster than the
speed of light. A frequently considered alternative is provided by the
telegraph equation, whose solutions avoid superluminal propagation speeds but
suffer from singular (non-continuous) diffusion fronts on the light cone, which
are unlikely to exist for massive particles. It is therefore advisable to
explore other alternatives as well. In this paper, a generalized Wiener process
is proposed that is continuous, avoids superluminal propagation, and reduces to
the standard Wiener process in the non-relativistic limit. The corresponding
relativistic diffusion propagator is obtained directly from the nonrelativistic
Wiener propagator, by rewriting the latter in terms of an integral over
actions. The resulting relativistic process is non-Markovian, in accordance
with the known fact that nontrivial continuous, relativistic Markov processes
in position space cannot exist. Hence, the proposed process defines a
consistent relativistic diffusion model for massive particles and provides a
viable alternative to the solutions of the telegraph equation.Comment: v3: final, shortened version to appear in Phys. Rev.
Analysis of Fourier transform valuation formulas and applications
The aim of this article is to provide a systematic analysis of the conditions
such that Fourier transform valuation formulas are valid in a general
framework; i.e. when the option has an arbitrary payoff function and depends on
the path of the asset price process. An interplay between the conditions on the
payoff function and the process arises naturally. We also extend these results
to the multi-dimensional case, and discuss the calculation of Greeks by Fourier
transform methods. As an application, we price options on the minimum of two
assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ
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