Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities

Consider a reflected random walk Wn+1=(Wn+Xn)+, where X0,X1,. are i.i.d. with negative mean and subexponential with common distribution F. It is shown that the probability that the maximum within a regenerative cycle with mean . exceeds x is approximately ..F(x) as x.., and thereby that max(W0,.,Wn) has the same asymptotics as max(X0,.,Xn) as n... In particular, the extremal index is shown to be .=0, and the point process of exceedances of a large level is studied. The analysis extends to reflected Lévy processes in continuous time, say, stable processes. Similar results are obtained for a storage process with release rate r(x) at level x and subexponential jumps (here the extremal index may be any value in [0,.]; also the tail of the stationary distribution is found. For a risk process with premium rate r(x) at level x and subexponential claims, the asymptotic form of the infinite-horizon ruin probability is determined. It is also shown by example [r(x)=a+bx and claims with a tail which is either regularly varying, Weibull- or lognormal-like] that this leads to approximations for finite-horizon ruin probabilities. Typically, the conditional distribution of the ruin time given eventual ruin is asymptotically exponential when properly normalized.