Let \((G_n)_{0\le n\le N}\) be a nonnegative process adapted to a filtration, with \(\mathrm{E}_P(G_n){\lt}\infty \) for every \(n\). Its Snell envelope \(S\), defined by backward induction, dominates \(G\), is a supermartingale under \(P\), and is dominated by every supermartingale that dominates \(G\). Moreover, \(\mathrm{E}_P(S_0)=\sup _\tau \mathrm{E}_P(G_{\tau \wedge N})\), where \(\tau \) ranges over the stopping times of the filtration.

Let \(\mathcal{X}\) be finite, let \(T\ge 0\), and let \(\xi \in \Delta _T\). Then \(\inf _{\bar P\in \mathcal{B}}\mathrm{KL}(Q^\xi \| \bar P)\le \sup _{(E_t)}\sum _{t=0}^T\xi _t\mathrm{E}_Q(\log E_t)\), where \((E_t)\) ranges over the e-processes. Their primal-dual equivalence, applied to a data-independent stopping time with distribution \(\xi \), gives the equality over their Bayesian mixtures of stopped null distributions, which belong to \(\mathcal{B}\).

For the random walk \((S_t)_{t\ge 0}\) with \(S_0=0\) and independent \(\mathrm{N}(0,1)\) increments, almost surely, \(\limsup _{t\to \infty }S_t/(2t\log \log t)^{1/2}=1\) and \(\liminf _{t\to \infty }S_t/(2t\log \log t)^{1/2}=-1\).

For every nonempty null of probability measures and every alternative probability measure, a numéraire exists at every horizon.

Let \(A(e)=(e-1-\log e)/(\log e)^2\) for \(e{\gt}1\) and \(A(1)=1/2\), the adjuster of their eq. (12). If \((M_t)\) is a test supermartingale with respect to a filtration coarser than \((\mathcal{F}_t)\), then the process with terms \(A(\max _{i\le t}M_i)\) is an e-process with respect to \((\mathcal{F}_t)\).