The Exact Ville Identity: From the Absorbing Case to the General Law with an Application to E-Values

Abstract

For a nonnegative supermartingale $(M_n)$ with $M_0=1$, let $T_b$ be the first time it reaches a level $b>1$. Ville's inequality gives only the bound $\mathbb P(T_b<\infty)\le 1/b$. We prove the exact identity $\mathbb P(T_b<\infty)=(1-D_b-R_b)/(b+O_b)$, where $O_b$ is the expected overshoot at crossing, $D_b$ is the cumulative predictable supermartingale loss before crossing, and $R_b=\lim_n\mathbb E[M_n{\bf 1}\{T_b>n\}]$ is the residual mass carried by paths that never cross. Thus the slack in Ville's inequality is decomposed completely into overshoot, loss, and survival. The proof is given first in the absorbing case, where non-crossing paths decay to zero and $R_b=0$, and then in full generality using a conservation identity for stopped martingales at extended-valued stopping times. The formula yields Ville's inequality as a corollary, gives a sharp tightness criterion, and is worked out in examples including double-or-absorb processes, multiplicative decay, gambler's ruin, likelihood-ratio martingales, and a bounded martingale with $R_b>0$. Finally, with $b=1/α$, the same identity gives the exact type-I error of a sequential e-value test and identifies when threshold recalibration can safely recover unused significance.