Jose Blanchet. Peter Glynn. “Complete corrected diffusion approximations for the maximum of a random walk.” Ann. Appl. Probab. 16 (2) 951 – 983, May 2006. https://doi.org/10.1214/105051606000000042

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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 …

Authors
Jose Blanchet, Peter Glynn
Publication date
2006/5/1
Volume
16
Issue
2
Pages
951-983