Exponential martingale of a biased simple random walk

For a walk with increments $+1$ with probability $p$ and $-1$ with probability $1-p$, putThen $\exp (\theta S_n-n\psi (\theta ))$ is a martingale. Optional stopping at the first linear-boundary crossing and optimization over $\theta$ gives exponential maximal-deviation bounds governed by the Legendre transform of a cumulant-generating function.