Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60F10 Large deviations
- 90A09 Finance, portfolios, investment
- 93E20 Optimal stochastic control

**Résumé:** We study an optimal investment model in which the goal is to maximize
the large deviation probability that the long term growth rate will
overperform a given target level. This criterion can be considered as
an asymptotic dynamic control version of the familiar Value
at Risk (VaR) concept. The optimal logarithmic moment gene\-rating
function is explicitly derived from an infinite time horizon risk
sensitive control problem. A careful study of its domain and its
behavior at the boundary of the domain is required. We then use
large deviations techniques for stating the optimal rate function
of the long term growth rate. This measure of upside-risk provides in
turn an objective probabilistic interpretation of the usually
subjective degree of risk aversion in CRRA utility function.

**Mots Clés:** *Large deviation ; risk sensitive control ; dynamic programming equation ; optimal logarithmic moment generating
function ; long term growth rate ; optimal portfolio*

**Date:** 2001-01-09

**Prépublication numéro:** *PMA-633*