25,069 research outputs found
Time dependence of entanglement entropy on the fuzzy sphere
We numerically study the behaviour of entanglement entropy for a free scalar
field on the noncommutative ("fuzzy") sphere after a mass quench. It is known
that the entanglement entropy before a quench violates the usual area law due
to the non-local nature of the theory. By comparing our results to the ordinary
sphere, we find results that, despite this non-locality, are compatible with
entanglement being spread by ballistic propagation of entangled quasi-particles
at a speed no greater than the speed of light. However, we also find that, when
the pre-quench mass is much larger than the inverse of the short-distance
cutoff of the fuzzy sphere (a regime with no commutative analogue), the
entanglement entropy spreads faster than allowed by a local model.Comment: 1+14 pages, 8 figures v2: References added, matches published versio
Mutual information on the fuzzy sphere
We numerically calculate entanglement entropy and mutual information for a
massive free scalar field on commutative (ordinary) and noncommutative (fuzzy)
spheres. We regularize the theory on the commutative geometry by discretizing
the polar coordinate, whereas the theory on the noncommutative geometry
naturally posseses a finite and adjustable number of degrees of freedom. Our
results show that the UV-divergent part of the entanglement entropy on a fuzzy
sphere does not follow an area law, while the entanglement entropy on a
commutative sphere does. Nonetheless, we find that mutual information (which is
UV-finite) is the same in both theories. This suggests that nonlocality at
short distances does not affect quantum correlations over large distances in a
free field theory.Comment: 16 pages, 10 figures. Fixed minor typos, references updated,
discussion slightly expande
Option pricing under fast-varying long-memory stochastic volatility
Recent empirical studies suggest that the volatility of an underlying price
process may have correlations that decay slowly under certain market
conditions. In this paper, the volatility is modeled as a stationary process
with long-range correlation properties in order to capture such a situation,
and we consider European option pricing. This means that the volatility process
is neither a Markov process nor a martingale. However, by exploiting the fact
that the price process is still a semimartingale and accordingly using the
martingale method, we can obtain an analytical expression for the option price
in the regime where the volatility process is fast mean-reverting. The
volatility process is modeled as a smooth and bounded function of a fractional
Ornstein-Uhlenbeck process. We give the expression for the implied volatility,
which has a fractional term structure
Correction to Black-Scholes formula due to fractional stochastic volatility
Empirical studies show that the volatility may exhibit correlations that
decay as a fractional power of the time offset. The paper presents a rigorous
analysis for the case when the stationary stochastic volatility model is
constructed in terms of a fractional Ornstein Uhlenbeck process to have such
correlations. It is shown how the associated implied volatility has a term
structure that is a function of maturity to a fractional power
Coupled paraxial wave equations in random media in the white-noise regime
In this paper the reflection and transmission of waves by a three-dimensional
random medium are studied in a white-noise and paraxial regime. The limit
system derives from the acoustic wave equations and is described by a coupled
system of random Schr\"{o}dinger equations driven by a Brownian field whose
covariance is determined by the two-point statistics of the fluctuations of the
random medium. For the reflected and transmitted fields the associated Wigner
distributions and the autocorrelation functions are determined by a closed
system of transport equations. The Wigner distribution is then used to describe
the enhanced backscattering phenomenon for the reflected field.Comment: Published in at http://dx.doi.org/10.1214/08-AAP543 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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