The Annals of Applied Probability

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

Isaac Meilijson

Full-text: Open access


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

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

First available in Project Euclid: 15 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Calmar ratio Crámer–Lundberg drawdown random walk Skorokhod embeddings


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.

Export citation


  • [1] Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability 2. World Scientific, River Edge, NJ.
  • [2] Atiya, A. F. and Magdon-Ismail, M. (2004). Maximum drawdown. Risk Magazine 17/10 99–102.
  • [3] Aumann, J. R. and Serrano, R. (2008). An economic index of riskiness. J. Political Economy 116 810–836.
  • [4] Azema, J. and Yor, M. (1978). a. Une solution simple au problème de Skorokhod. b. Le problème de Skorokhod: Compléments. In Séminaire de Probabilités XIII. Lecture Notes in Math. 721 90–115, 625–633. Springer, Berlin.,.
  • [5] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [6] Blanchet, J. and Glynn, P. I. (2004). Complete corrected diffusion approximations for the maximum of a random walk. Unpublished manuscript.
  • [7] Chacon, R. V. and Walsh, J. B. (1976). One-dimensional potential embedding. In Séminaire de Probabilités, X (Prèmiere Partie, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975). Lecture Notes in Math. 511 19–23. Springer, Berlin.
  • [8] Chang, J. T. and Peres, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Probab. 25 787–802.
  • [9] Duadi, R., Shiryaev, A. N. and Yor, M. (1999). On the probability characteristics of “drop” variables in standard Brownian motion. Teor. Veroyatnost. i Primenen. 44 3–13.
  • [10] Dubins, L. E. (1968). On a theorem of Skorohod. Ann. Math. Statist. 39 2094–2097.
  • [11] Dubins, L. E. and Gilat, D. (1978). On the distribution of maxima of martingales. Proc. Amer. Math. Soc. 68 337–338.
  • [12] Dubins, L. E. and Schwarz, G. (1988). A sharp inequality for sub-martingales and stopping-times. Astérisque 157–158 129–145.
  • [13] Goldhirsch, I. and Noskovicz, S. H. (1990). The first passage time distribution in random walk. Phys. Rev. A 42 2047–2064.
  • [14] Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
  • [15] Holewijn, P. J. and Meilijson, I. (1983). Note on the central limit theorem for stationary processes. In Seminar on Probability, XVII. Lecture Notes in Math. 986 240–242. Springer, Berlin.
  • [16] Kingman, J. F. C. (1963). Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. Lond. Math. Soc. 13 593–604.
  • [17] Meilijson, I. (1983). On the Azéma–Yor stopping time. In Seminar on Probability, XVII. Lecture Notes in Math. 986 225–226. Springer, Berlin.
  • [18] Meilijson, I. (2003). The time to a given drawdown in Brownian motion. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 94–108. Springer, Berlin.
  • [19] Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems. Adv. in Appl. Probab. 11 701–719.
  • [20] Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.
  • [21] Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Probab. 3 234–246.
  • [22] Wald, A. (1947). Sequential Analysis. Wiley, New York.