The Annals of Applied Probability

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

Hiroaki Hata, Hideo Nagai, and Shuenn-Jyi Sheu

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

First available in Project Euclid: 8 January 2010

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Primary: 35J60: Nonlinear elliptic equations 49L20: Dynamic programming method 60F10: Large deviations 91B28 93E20: Optimal stochastic control

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


Hata, Hiroaki; Nagai, Hideo; Sheu, Shuenn-Jyi. Asymptotics of the probability minimizing a “down-side” risk. Ann. Appl. Probab. 20 (2010), no. 1, 52--89. doi:10.1214/09-AAP618.

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