Complete corrected diffusion approximations for the maximum of a random walk

Consider a random walk (Sn:n.0) with drift .. 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 .>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 .) admits an analytic extension throughout a neighborhood of the origin in the complex plane .. Finally, when the increments of the random walk have nonnegative mean ., we show that the Laplace transform, E.exp(.bR(.)), of the limiting overshoot, R(.), can be analytically extended throughout a disc centered at the origin in . . . (jointly for both b and .). In addition, when the distribution of the increments is continuous and appropriately symmetric, we show that E.S. [where . is the first (strict) ascending ladder epoch] can be analytically extended to a disc centered at the origin in ., 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].