17,770 research outputs found
On matrix elements for the quantized cat map modulo prime powers
The quantum cat map is a model for a quantum system with underlying chaotic
dynamics. In this paper we study the matrix elements of smooth observables in
this model, when taking arithmetic symmetries into account. We give explicit
formulas for the matrix elements as certain exponential sums. With these
formulas we can show that there are sequences of eigenfunctions for which the
matrix elements decay significantly slower then was previously conjectured. We
also prove a limiting distribution for the fluctuation of the normalized matrix
elements around their average.Comment: 26 pages, final version, to appear in AH
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
Regenerative Composition Structures
A new class of random composition structures (the ordered analog of Kingman's
partition structures) is defined by a regenerative description of component
sizes. Each regenerative composition structure is represented by a process of
random sampling of points from an exponential distribution on the positive
halfline, and separating the points into clusters by an independent
regenerative random set. Examples are composition structures derived from
residual allocation models, including one associated with the Ewens sampling
formula, and composition structures derived from the zero set of a Brownian
motion or Bessel process. We provide characterisation results and formulas
relating the distribution of the regenerative composition to the L{\'e}vy
parameters of a subordinator whose range is the corresponding regenerative set.
In particular, the only reversible regenerative composition structures are
those associated with the interval partition of generated by excursions
of a standard Bessel bridge of dimension for some
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