Asymptotics for the expected maximum of random walks and Levy flights with a constant drift

In this paper, we study the large n asymptotics of the expected maximum of an n-step random walk/Levy flight (characterized by a Levy index 1 < mu <= 2) on a line, in the presence of a constant drift c. For 0 < mu <= 1, the expected maximum is infinite, even for finite values of n. For 1 < mu <= 2, we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large n. For c < 0 and mu = 2, the expected maximum approaches a non-trivial constant as n gets large, while for 1 < mu < 2, it grows as a power law similar to n(2-mu). For c > 0, the asymptotic expansion of the expected maximum is simply related to the one for c < 0 by adding to the latter the linear drift term en, making the leading term grow linearly for large n, as expected. Finally, we derive a scaling form interpolating smoothly between the cases c = 0 and c not equal 0. These results are borne out by numerical simulations in excellent agreement with our analytical predictions.