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In this paper, we study the large $n$ asymptotics of the expected maximum ofan $n$-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 asymptoticexpansion of the expected maximum for large $n$. For $c<0$ and $\mu =2$, theexpected 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 $c>0$, theasymptotic 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 leadingterm grow linearly for large $n$, as expected. Finally, we derive a scalingform interpolating smoothly between the cases $c=0$ and $c\ne 0$. These resultsare borne out by numerical simulations in excellent agreement with ouranalytical predictions.

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