Ecole Polytechnique

In this talk, we will focus on the asymptotic behavior in time of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time.

The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude tends to zero.

The initial condition is a standing wave solution of the unperturbed equation We prove that up to times of the order of the inverse squared amplitude the solution decomposes into the sum of a randomly modulatedmodulation parameters.

In addition, we show that the first order of the remainder, as the noise amplitude goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale, as the frequency of the standing wave of the deterministic equation tends to its minimal value.

We study the homogenization and singular perturbation of the

wave equation in a periodic media for long times of the order

of the inverse of the period. We consider inital data that are

Bloch wave packets, i.e., that are the product of a fast

oscillating Bloch wave and of a smooth envelope function.

We prove that the solution is approximately equal to two waves

propagating in opposite directions at a high group velocity with

envelope functions which obey a Schr\"{o}dinger type equation.

Our analysis extends the usual WKB approximation by adding a

dispersive, or diffractive, effect due to the non uniformity

of the group velocity which yields the dispersion tensor of

the homogenized Schr\"{o}dinger equation. This is a joint

work with M. Palombaro and J. Rauch.

Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints.

The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.

We discuss estimating the growth exponents for positive solutions to the

random parabolic Anderson's model with small parameter k. We show that

behaviour for the case where the spatial variable is continuous differs

markedly from that for the discrete case.