The Maximum on a Random Time Interval of a Random Walk with Long-tailed Increments and Negative Drift
نویسنده
چکیده
Random walks with long-tailed increments have many important applications in insurance, finance, queueing networks, storage processes, and the study of extreme events in nature and elsewhere. See, for example, Embrechts et al. (1997), Asmussen (1998, 1999) and Greiner et al. (1999) for some background. In this paper we study the distribution of the maximum of such a random walk over a random time interval. Let F be the distribution function of the increments of a random walk {Sn}n≥0 with S0 = 0. Suppose that this distribution has a finite negative mean and that F is longtailed in the positive direction (see below for this and other definitions). Of interest is the asymptotic distribution of the maximum of {Sn} over the interval [0, σ] defined by some stopping time σ. Some results for the case where σ is independent of {Sn} are known (again see below). However, relatively little is known for other stopping times. Asmussen (1998) gives the expected result for the case σ = τ , where
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