73 research outputs found
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
A general framework for waves in random media with long-range correlations
We consider waves propagating in a randomly layered medium with long-range
correlations. An example of such a medium is studied in \citeMS and leads, in
particular, to an asymptotic travel time described in terms of a fractional
Brownian motion. Here we study the asymptotic transmitted pulse under very
general assumptions on the long-range correlations. In the framework that we
introduce in this paper, we prove in particular that the asymptotic time-shift
can be described in terms of non-Gaussian and/or multifractal processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP689 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Retrieval of a Green's function with reflections from partly coherent waves generated by a wave packet using cross correlations
International audienc
On effective attenuation in multiscale composite media
International audienc
Optimal Hedging Under Fast-Varying Stochastic Volatility
International audienceIn a market with a rough or Markovian mean-reverting stochastic volatility thereis no perfect hedge. Here it is shown how various delta-type hedging strategies perform and canbe evaluated in such markets in the case of European options.A precise characterization of thehedging cost, the replication cost caused by the volatilityfluctuations, is presented in an asymptoticregime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset basedhedging strategy in the considered regime is identified as the so-called “practitioners” delta hedgingscheme. It is moreover shown that the performances of the delta-type hedging schemes are essentiallyindependent of the regularity of the volatility paths in theconsidered regime and that the hedgingcosts are related to a Vega risk martingale whose magnitude is proportional to a new market riskparameter. It is also shown via numerical simulations that the proposed hedging schemes whichderive from option price approximations in the regime of rapid mean reversion, are robust: the“practitioners” delta hedging scheme that is identified as being optimal by our asymptotic analysiswhen the mean reversion time is small seems to be optimal witharbitrary mean reversion times
Wave Propagation in Random Waveguides with Long-Range Correlations
International audienceThe paper presents an analysis of acoustic wave propagation in a waveguide with random fluctuations of its sound speed profile. These random perturbations are assumed to have long-range correlation properties. In the waveguide, a monochromatic wave can be decomposed in propagating modes and evanescent modes, and the random perturbation couples all these modes. The paper presents an asymptotic analysis of the mode-coupling mechanism and uses this to characterize the transmitted wave. The paper presents the first fully rigorous characterization of wave propagation in long-range non-layered media
A Two-way Paraxial System for Simulation of Wave Backscattering by a Random Medium
This paper presents a probabilistic analysis of an iterative two-way paraxial scheme for the simulation of wave propagation in random media. This scheme has the computational cost of the standard one-way paraxial wave equation but has the accuracy of the full wave equation in a regime beyond the classical paraxial regime. More precisely, it accurately predicts the statistics of the reflected wave field. The accuracy is determined by the order of the iterative scheme and the ratio of the random backscattering intensity over the random forward-scattering intensity
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