Asymptotics for the Expected Maximum of Random Walks and L\'evy Flights with a Constant Drift

In this paper, we study the large $n$ asymptotics of the expected maximum of an $n$-step random walk/L\'evy flight (characterized by a L\'evy index $1<\mu\leq 2$) on a line, in the presence of a constant drift $c$. For $0<\mu\leq 1$, the expected maximum is infinite, even for finite values of $n$. For $1<\mu\leq 2$, 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 $\sim n^{2-\mu}$. 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 $cn$, 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 $c\ne 0$. These results are borne out by numerical simulations in excellent agreement with our analytical predictions.