dc.contributor.author Espinosa, Gilles-Edouard dc.contributor.author Elie, Romuald dc.date.accessioned 2011-03-07T11:46:20Z dc.date.available 2011-03-07T11:46:20Z dc.date.issued 2011-02 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/5730 dc.language.iso en en dc.subject Mean reverting diffusion en dc.subject Verification en dc.subject Free boundary PDE en dc.subject Running maximum en dc.subject Optimal prediction en dc.subject Optimal stopping en dc.subject.ddc 519 en dc.title Optimal stopping of a mean reverting diffusion: minimizing the relative distance to the maximum en dc.type Document de travail / Working paper dc.contributor.editoruniversityother Departement of Mathematics, ETH Zurich;Suisse dc.contributor.editoruniversityother Centre de Recherche en Économie et Statistique (CREST) INSEE – École Nationale de la Statistique et de l'Administration Économique;France dc.description.abstracten 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. en dc.publisher.name Université Paris-Dauphine en dc.publisher.city Paris en dc.identifier.citationpages 37 en dc.identifier.urlsite http://hal.archives-ouvertes.fr/hal-00573429/fr/ en dc.description.sponsorshipprivate oui en dc.subject.ddclabel Probabilités et mathématiques appliquées en
﻿

## Files in this item

FilesSizeFormatView

There are no files associated with this item.