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

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

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

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