99 research outputs found
On the probability of hitting the boundary for Brownian motions on the SABR plane
Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods
Implied volatility of basket options at extreme strikes
In the paper, we characterize the asymptotic behavior of the implied
volatility of a basket call option at large and small strikes in a variety of
settings with increasing generality. First, we obtain an asymptotic formula
with an error bound for the left wing of the implied volatility, under the
assumption that the dynamics of asset prices are described by the
multidimensional Black-Scholes model. Next, we find the leading term of
asymptotics of the implied volatility in the case where the asset prices follow
the multidimensional Black-Scholes model with time change by an independent
increasing stochastic process. Finally, we deal with a general situation in
which the dependence between the assets is described by a given copula
function. In this setting, we obtain a model-free tail-wing formula that links
the implied volatility to a special characteristic of the copula called the
weak lower tail dependence function
Mean Dimension of Function Classes with Lebesgue Measurable Spectral Sets
AbstractThe notion of mean dimension was introduced in the 1970s by Tikhomirov. It determines the mean number of linear dimensions required to identify an element of a given function class. Tikhomirov then posed the following problem: find the mean dimension of the unit ball BpE of the space of Lp-functions on Rn with spectra inside a given Lebesgue measurable bounded set E. In the language of signal analysis: determine the amount of linear information carried by generalized band-limited signals. In this paper Tikhomirov′s conjecture on mean dimension is confirmed in certain important cases and yet shown to fail in certain other cases
Rearrangements of Functions on a Locally Compact Abelian Group and Integrability of the Fourier Transform
AbstractWe find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group. These classes consist of all functions for which the Fourier transform belongs to a given Lorentz space on the dual group. Different special cases of the problems considered in this paper have been originally studied by Hardy, Littlewood, Hewitt, Ross, Cereteli, and the author
Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
We prove local and global well-posedness for semi-relativistic, nonlinear
Schr\"odinger equations with
initial data in , . Here is a critical
Hartree nonlinearity that corresponds to Coulomb or Yukawa type
self-interactions. For focusing , which arise in the quantum theory of
boson stars, we derive a sufficient condition for global-in-time existence in
terms of a solitary wave ground state. Our proof of well-posedness does not
rely on Strichartz type estimates, and it enables us to add external potentials
of a general class.Comment: 18 pages; replaced with revised version; remark and reference on blow
up adde
Dynamical Collapse of Boson Stars
We study the time evolution in system of bosons with a relativistic
dispersion law interacting through an attractive Coulomb potential with
coupling constant . We consider the mean field scaling where tends to
infinity, tends to zero and remains fixed. We investigate
the relation between the many body quantum dynamics governed by the
Schr\"odinger equation and the effective evolution described by a
(semi-relativistic) Hartree equation. In particular, we are interested in the
super-critical regime of large (the sub-critical case has been
studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have
solutions which blow up in finite time. To inspect this regime, we need to
regularize the Coulomb interaction in the many body Hamiltonian with an
dependent cutoff that vanishes in the limit . We show, first, that
if the solution of the nonlinear equation does not blow up in the time interval
, then the many body Schr\"odinger dynamics (on the level of the
reduced density matrices) can be approximated by the nonlinear Hartree
dynamics, just as in the sub-critical regime. Moreover, we prove that if the
solution of the nonlinear Hartree equation blows up at time (in the sense
that the norm of the solution diverges as time approaches ), then
also the solution of the linear Schr\"odinger equation collapses (in the sense
that the kinetic energy per particle diverges) if and,
simultaneously, sufficiently fast. This gives the first
dynamical description of the phenomenon of gravitational collapse as observed
directly on the many body level.Comment: 40 page
Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
We consider the semi-relativistic system of gravitating Bosons with
gravitation constant . The time evolution of the system is described by the
relativistic dispersion law, and we assume the mean-field scaling of the
interaction where and while fixed. In
the super-critical regime of large , we introduce the regularized
interaction where the cutoff vanishes as . We show that the
difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and
the corresponding semi-relativistic Hartree dynamics is at most of order
for all , i.e., the result covers the sub-critical regime and
the super-critical regime. The dependence of the bound is optimal.Comment: 29 page
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