Asymptotics of the probability minimizing a “down-side” risk

Asymptotics of the probability minimizing a “down-side” risk

Hiroaki Hata, Hideo Nagai, and Shuenn-Jyi Sheu

Source: Ann. Appl. Probab. Volume 20, Number 1 (2010), 52-89.

Abstract

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.

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Primary Subjects: 35J60, 49L20, 60F10, 91B28, 93E20

Keywords: Large deviation; long-term investment; risk-sensitive stochastic control; Bellman equation

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1262962318
Digital Object Identifier: doi:10.1214/09-AAP618
Zentralblatt MATH identifier: 05678699
Mathematical Reviews number (MathSciNet): MR2582642

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