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&#39;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&#39;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&#39;s inequality as a corollary, gives a sharp tightness criterion, and is worked out in examples including double-or-absorb processes, multiplicative decay, gambler&#39;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.