The Annals of Applied Probability

Complete corrected diffusion approximations for the maximum of a random walk

Jose Blanchet and Peter Glynn

Full-text: Open access

Abstract

Consider a random walk (Sn:n≥0) with drift −μ and S0=0. Assuming that the increments have exponential moments, negative mean, and are strongly nonlattice, we provide a complete asymptotic expansion (in powers of μ>0) that corrects the diffusion approximation of the all time maximum M=maxn≥0Sn. Our results extend both the first-order correction of Siegmund [Adv. in Appl. Probab. 11 (1979) 701–719] and the full asymptotic expansion provided in the Gaussian case by Chang and Peres [Ann. Probab. 25 (1997) 787–802]. We also show that the Cramér–Lundberg constant (as a function of μ) admits an analytic extension throughout a neighborhood of the origin in the complex plane ℂ. Finally, when the increments of the random walk have nonnegative mean μ, we show that the Laplace transform, Eμexp(−bR(∞)), of the limiting overshoot, R(∞), can be analytically extended throughout a disc centered at the origin in ℂ × ℂ (jointly for both b and μ). In addition, when the distribution of the increments is continuous and appropriately symmetric, we show that EμSτ [where τ is the first (strict) ascending ladder epoch] can be analytically extended to a disc centered at the origin in ℂ, generalizing the main result in [Ann. Probab. 25 (1997) 787–802] and extending a related result of Chang [Ann. Appl. Probab. 2 (1992) 714–738].

Article information

Source
Ann. Appl. Probab. Volume 16, Number 2 (2006), 951-983.

Dates
First available in Project Euclid: 29 June 2006

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1151592256

Digital Object Identifier
doi:10.1214/105051606000000042

Mathematical Reviews number (MathSciNet)
MR2244438

Zentralblatt MATH identifier
1132.60038

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems 62L10: Sequential analysis 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Keywords
Corrected diffusion approximations random walks ladder heights sequential analysis single-server queue

Citation

Blanchet, Jose; Glynn, Peter. Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Probab. 16 (2006), no. 2, 951--983. doi:10.1214/105051606000000042. http://projecteuclid.org/euclid.aoap/1151592256.


Export citation

References

  • Asmussen, S. (2001). Ruin Probabilities. World Scientific, Singapore.
  • Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.
  • Broadie, M., Glasserman, P. and Kou, S. (1997). A continuity correction for discrete barrier options. Math. Finance 7 325--349.
  • Breiman, L. (1992). Probability. Addison--Wesley, Reading, MA.
  • Butzer, P. and Nessel, R. (1971). Fourier Analysis and Approximation 1. Birkhäuser, Boston.
  • Chang, J. (1992). On moments of the first ladder height of random walks with small drift. Ann. Appl. Probab. 2 714--738.
  • Chang, J. and Peres, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Probab. 25 787--802.
  • Glasserman, P. and Liu, T. (1997). Corrected diffusion approximations for multistage production-inventory systems. Math. Oper. Res. 12 186--201.
  • Kiefer, J. and Wolfowitz, J. (1956). On the characteristics of the general queueing process, with applications to random walk. Ann. Math. Statist. 27 147--161.
  • Kingman, J. (1963). Ergodic properties of continuous time Markov processes and their discrete skeletons. Proc. London. Math. Soc. 13 593--604.
  • Lai, T. (1976). Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab. 4 51--66.
  • Lindley, D. (1952). The theory of a queue with a single-server. Proc. Cambridge Philos. Soc. 48 277--289.
  • Lotov, V. (1996). On some boundary crossing problems for Gaussian random walks. Ann. Probab. 24 2154--2171.
  • Rudin, W. (1987). Real and Complex Analysis. McGraw--Hill, New York.
  • Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems. Adv. in Appl. Probab. 11 701--719.
  • Siegmund, D. (1985). Sequential Analysis. Springer, New York.
  • Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika 66 453--463.