Open Access
June 2009 On the adjustment coefficient, drawdowns and Lundberg-type bounds for random walk
Isaac Meilijson
Ann. Appl. Probab. 19(3): 1015-1025 (June 2009). DOI: 10.1214/08-AAP567


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.


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Isaac Meilijson. "On the adjustment coefficient, drawdowns and Lundberg-type bounds for random walk." Ann. Appl. Probab. 19 (3) 1015 - 1025, June 2009.


Published: June 2009
First available in Project Euclid: 15 June 2009

zbMATH: 1235.60048
MathSciNet: MR2537197
Digital Object Identifier: 10.1214/08-AAP567

Primary: 60G44 , 60G50
Secondary: 91B30

Keywords: Calmar ratio , Crámer–Lundberg , Drawdown , Random walk , Skorokhod embeddings

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.19 • No. 3 • June 2009
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