Résumé (EN)

Considering a diffusion $X$ mean reverting to 0 {and starting at $X_0>0$}, we study the control problem $$ \inf_\theta \Esp{f\left(\frac{X_\theta}{\Sup_{s\in[0,\tau]}{X_s}}\right)}\;,$$ where $f$ is a given function and $\tau$ is the next random time where the diffusion $X$ crosses zero. Our motivation is the obtention of optimal selling rules related to the minimization of the relative distance between a stopped mean reverting portfolio and its upcoming maximum. We provide a verification result for this stochastic control problem and derive the solution for different criteria $f$. For a power utility type criterion $f:y \mapsto - {y^\la}$ with $\la>0$, instantaneous stopping is always optimal. On the contrary, for a relative quadratic error criterion, $f:y \mapsto {(1-y)^2}$, selling is optimal as soon as the process $X$ crosses a specified function $\varphi$ of its running maximum $X^*$. As in [5] and [8], the inverse of $\varphi$ identifies as the maximal solution of a highly non linear ordinary differential equation. These results reinforce the idea that optimal prediction problems of similar type lead easily to solutions of different nature. Nevertheless, we observe numerically that the continuation region for the relative quadratic error criterion is very small, so that the optimal selling strategy is close to immediate stopping.