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

TL;DR

This paper derives bounds for the expected maximum of a light-tailed random walk until a specified drawdown, connecting concepts from risk theory and stochastic processes using advanced martingale techniques.

Contribution

It introduces new bounds for the maximum of a random walk until a drawdown, extending Lundberg bounds with martingale embeddings and linking to riskiness indices.

Findings

Bounds for expected maximum involving the adjustment coefficient alpha

Exponential stochastic upper and lower bounds for the minimum of the walk

Tail probability bounds of the form C exp{-alpha x}

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{alpha d}-1)/alpha and (K exp{alpha d}-1)/alpha for some K>1, in terms of the adjustment coefficient alpha (E[exp{-alpha X}]=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 Cramer-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{-alpha x} and exp{-alpha x} respectively, for some 1/K < C < 1. Our treatment of the problem involves Skorokhod embeddings of random walks in Martingales, especially via the Azema-Yor and Dubins stopping times, adapted from standard Brownian Motion to exponential Martingales.