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

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 (K exp{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 C exp{−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.