Complete corrected diffusion approximations for the maximum of a random walk

Abstract

Consider a random walk (S_n: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=max_n≥0S_n. 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, E_μexp(−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 E_μS_τ [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].