Ville's inequality for nonnegative test supermartingales

Fix any $\gamma \in (0,1)$ and define ${\tau }_{\gamma }=\inf \{t\ge 0:E_t\ge 1\mathrm{/}\gamma \}$, with $\inf \mathrm{\varnothing }=\mathrm{\infty }$. For every $n$, $\{{\tau }_{\gamma }\le n\}={\bigcup }_{k=0}^n\{E_k\ge 1\mathrm{/}\gamma \}\in {\mathcal{F}}_n$, so ${\tau }_{\gamma }$ is a stopping time. Also $\{{\tau }_{\gamma }\wedge N\le n\}=\{{\tau }_{\gamma }\le n\}$ for $n<N$ and is the whole space for $n\ge N$, so each ${\tau }_{\gamma }\wedge N$ is a bounded stopping time.

The constant time $0$ is a stopping time bounded by $0$, and step 1.1 shows ${\tau }_{\gamma }\wedge N$ is a bounded stopping time. Bounded optional sampling therefore gives ${\mathbb{E}}_P{E}_{{\tau }_{\gamma }\wedge N}\le {\mathbb{E}}_P{E}_0\le 1$.

On $\{{\tau }_{\gamma }\le N\}$, $E_{{\tau }_{\gamma }\wedge N}\ge 1\mathrm{/}\gamma$; on its complement the stopped value is $E_N\ge 0$ almost surely. Hence ${\mathbb{E}}_P{E}_{{\tau }_{\gamma }\wedge N}\ge {\gamma }^{-1}P({\tau }_{\gamma }\le N)$, and step 2.1 yields $P({\tau }_{\gamma }\le N)\le \gamma$.

The events $\{{\tau }_{\gamma }\le N\}$ increase with $N$ and their union is $\{{\tau }_{\gamma }<\mathrm{\infty }\}={\bigcup }_{t\ge 0}\{E_t\ge 1\mathrm{/}\gamma \}$. Continuity from below and step 3.1 therefore give $P(\mathrm{\exists }t\ge 0:E_t\ge 1\mathrm{/}\gamma )\le \gamma$.

Let $H=\{{\sup }_{t\ge 0}{E}_t\ge 1\mathrm{/}\alpha \}$ and suppose $q=P(H)>\alpha$. Since $q\le 1$, $\beta =(\alpha +q)\mathrm{/}2$ lies in $(\alpha ,1)$, so $1\mathrm{/}\beta <1\mathrm{/}\alpha$. If every $E_t\le 1\mathrm{/}\beta$, then ${\sup }_t{E}_t\le 1\mathrm{/}\beta <1\mathrm{/}\alpha$, contrary to $H$; hence $H\subseteq \{\mathrm{\exists }t:E_t\ge 1\mathrm{/}\beta \}$ even if the supremum is not attained. Step 4.1 with $\gamma =\beta$ gives $q\le \beta <q$, a contradiction. Therefore $P(H)\le \alpha$, and since $P$ was arbitrary, the bound holds under every null law.