Ecole Polytechnique

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.