Constrained portfolio optimisation via martingale techniques: on Azema- Yor processes as solutions to SDEs.

I consider the problem of maximising the final utility of a portfolio which is constrained to satisfy the draw-down condition, i.e. the current value of the portfolio can not drop below a pre-specified funciton of its running maximum. It turns out that martingale techniques yield an explicit and rather elegant solution. The so- called Azema-Yor processes appear naturally and I take some time to introduce this class and discuss some of their remarkable properties.

In particular, I show how they can be characterised as (unique,

strong) solutions to SDEs called the Bachelier Eq and the Draw-Down Eq.

The talk is based (in particular) on a joint work with L. Carraro, N.

El Karoui and A. Meziou.