The Annals of Applied Probability

On the adjustment coefficient, drawdowns and Lundberg-type bounds for random walk

Isaac Meilijson

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Abstract

Consider a random walk whose (light-tailed) increments have positive mean. Lower and upper bounds are provided for the expected maximal value of the random walk until it experiences a given drawdown d. These bounds, related to the Calmar ratio in finance, are of the form (exp{αd}−1)/α and (Kexp{αd}−1)/α for some K>1, in terms of the adjustment coefficient α (E[exp{−αX}]=1) of the insurance risk literature. Its inverse 1/alpha has been recently derived by Aumann and Serrano as an index of riskiness of the random variable X.

This article also complements the Lundberg exponential stochastic upper bound and the Crámer–Lundberg approximation for the expected minimum of the random walk, with an exponential stochastic lower bound. The tail probability bounds are of the form Cexp{−αx} and exp{−αx}, respectively, for some 1/K<C<1.

Our treatment of the problem involves Skorokhod embeddings of random walks in martingales, especially via the Azéma–Yor and Dubins stopping times, adapted from standard Brownian motion to exponential martingales.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 3 (2009), 1015-1025.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071017

Digital Object Identifier
doi:10.1214/08-AAP567

Mathematical Reviews number (MathSciNet)
MR2537197

Zentralblatt MATH identifier
1235.60048

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60G44: Martingales with continuous parameter
Secondary: 91B30: Risk theory, insurance

Keywords
Calmar ratio Crámer–Lundberg drawdown random walk Skorokhod embeddings

Citation

Meilijson, Isaac. On the adjustment coefficient, drawdowns and Lundberg-type bounds for random walk. Ann. Appl. Probab. 19 (2009), no. 3, 1015--1025. doi:10.1214/08-AAP567. https://projecteuclid.org/euclid.aoap/1245071017


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