The Annals of Applied Probability

Downside risk minimization via a large deviations approach

Hideo Nagai

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We consider minimizing the probability of falling below a target growth rate of the wealth process up to a time horizon T in an incomplete market model, and then study the asymptotic behavior of minimizing probability as T → ∞. This problem can be closely related to an ergodic risk-sensitive stochastic control problem in the risk-averse case. Indeed, in our main theorem, we relate the former problem concerning the asymptotics for risk minimization to the latter as its dual. As a result, we obtain an expression of the limit value of the probability as the Legendre transform of the value of the control problem, which is characterized as the solution to an H-J-B equation of ergodic type, in the case of a Markovian incomplete market model.

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Ann. Appl. Probab., Volume 22, Number 2 (2012), 608-669.

First available in Project Euclid: 2 April 2012

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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 H-J-B equation of ergodic type


Nagai, Hideo. Downside risk minimization via a large deviations approach. Ann. Appl. Probab. 22 (2012), no. 2, 608--669. doi:10.1214/11-AAP781.

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