2 The setting
Throughout this chapter, the data \(X=(X_t)_{t\ge 1}\) take values in a measurable space \(\mathcal{X}\), the null \(\mathcal{P}\) is a set of distributions of \(X\), and \(Q\) is an alternative distribution of \(X\). These are the results of Sections 1 and 2 of the paper and of Sections S1.1 and S1.2, and the general claims of Example S1 and Section S4.
The filtration \((\mathcal{F}_t)_{t\ge 0}\) is generated by \(X\): the \(\sigma \)-algebra \(\mathcal{F}_t\) is generated by \(X^t=(X_1,\ldots ,X_t)\), and \(\mathcal{F}_0\) is trivial. For a distribution \(P\) of \(X\), \(P_t\) is the distribution of \(X^t\).
An e-variable at horizon \(t\) is an \(\mathcal{F}_t\)-measurable random variable \(E\) with values in \([0,\infty ]\) such that \(\sup _{P\in \mathcal{P}}\mathrm{E}_P(E)\le 1\), and \(\mathcal{E}_t\) is the set of these e-variables. An e-process is a process \((E_t)_{t\ge 0}\) with values in \([0,\infty ]\), adapted to \((\mathcal{F}_t)\), such that \(\sup _{P\in \mathcal{P}}\mathrm{E}_P(E_\tau )\le 1\) for every stopping time \(\tau \) of \((\mathcal{F}_t)\) with values in \(\{ 0,1,\ldots \} \cup \{ \infty \} \), where \(E_\infty =\limsup _{t\to \infty }E_t\).
A test supermartingale, or test martingale, with respect to a filtration is a nonnegative process adapted to this filtration, with initial value one, that is a supermartingale, or a martingale, with respect to this filtration under every null distribution.
A numéraire at horizon \(t\) is an e-variable \(E^{\star }_t\in \mathcal{E}_t\) such that \(\mathrm{E}_Q(E/E^{\star }_t)\le 1\) for every \(E\in \mathcal{E}_t\) (Larsson, Ramdas and Ruf, 2025, Definition 2.1). The e-power of \(E\) is \(\mathrm{E}_Q(\log E)\). The numéraire is log-optimal if \(\mathrm{E}_Q(\log E^{\star }_t)=\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E)\), and \(E^{\star }_t=\mathrm{d}Q_t/\mathrm{d}P^{\star }_t\) if \(Q_t\) has the density \(e\) with respect to a sub-probability measure \(P^{\star }_t\) on \(\mathcal{X}^t\), with \(E^{\star }_t=e(X^t)\) \(Q\)-almost surely.
The expectation of a random variable with values in \([-\infty ,\infty ]\) is the integral of its positive part minus that of its negative part. The regret of \(E\in \mathcal{E}_t\) is \(R_t(E)=\mathrm{E}_Q\{ \log (E^{\star }_t/E)\} \), the regret of an e-process \((E_t)\) at \(t\) is \(R_t(E_t)\), and the minimax regret is \(\Gamma _T=\inf _{(E_t)}\max _{0\le t\le T}R_t(E_t)\), where \((E_t)\) ranges over the e-processes.
The null distributions and \(Q\) are probability measures, \(E^{\star }_t\) is a numéraire at every horizon \(t\), in versions fixed once and for all, and \(\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E){\lt}\infty \) for every \(t\ge 0\).
Let
where \(\tau \) ranges over the stopping times of \((\mathcal{F}_t)\). For \(P\in \mathcal{P}\), the Snell envelope of \((E^{\star }_{t\wedge T})_{t\ge 0}\) under \(P\) is defined by \(\Lambda ^P_T=E^{\star }_T\) and
The set \(\Delta _T\) contains the probability vectors \((\xi _0,\ldots ,\xi _T)\).
For a probability measure \(\mu \) and a measurable \(f\) with values in \([0,\infty ]\), \(\mathrm{E}_\mu (\log f)\le \log \mathrm{E}_\mu (f)\).
The tangent line \(\log x\le x-1\), applied to \(f/\mathrm{E}_\mu (f)\).
For every \(t\ge 0\), \(E^{\star }_t\) is positive and finite \(Q\)-almost surely, \(\log E^{\star }_t\) is \(Q\)-integrable, and \(E^{\star }_t\) is log-optimal. For every \(E\in \mathcal{E}_t\), the negative part of \(\log (E^{\star }_t/E)\) is \(Q\)-integrable, so that \(R_t(E)\) is well defined, and \(R_t(E)=\mathrm{E}_Q(\log E^{\star }_t)-\mathrm{E}_Q(\log E)\in [0,\infty ]\).
As in Section S1.1: the numéraire property and Jensen’s inequality for the logarithm give \(\mathrm{E}_Q\{ \log (E/E^{\star }_t)\} \le \log \mathrm{E}_Q(E/E^{\star }_t)\le 0\) for every \(E\in \mathcal{E}_t\), and the standing assumption bounds the e-power of \(E^{\star }_t\) from above.
The numéraire \(E^{\star }_t\) is positive \(Q\)-almost surely and unique up to \(Q\)-null sets, and \(E^{\star }_0=1\) since \(\mathcal{F}_0\) is trivial (Section 2).
For two numéraires \(E_1\) and \(E_2\), the ratio \(U=E_2/E_1\) satisfies \(\mathrm{E}_Q(U)\le 1\) and \(\mathrm{E}_Q(1/U)\le 1\). Since \(U+1/U\ge 2\), with equality only at \(U=1\), \(U=1\) \(Q\)-almost surely; this replaces the equality case of Jensen’s inequality in Larsson, Ramdas and Ruf (2025, Section 2). An \(\mathcal{F}_0\)-measurable numéraire is a constant \(c\) with \(c\le 1\) and \(1/c\le 1\), which needs a nonempty null, given by the standing assumption.
Test supermartingales with respect to \((\mathcal{F}_t)\) are e-processes (Section 1).
Optional stopping at the bounded times \(\tau \wedge t\) and Fatou’s lemma. On \(\{ \tau =\infty \} \), Fatou’s lemma bounds the limit inferior, while \(E_\infty \) is the limit superior; Doob’s convergence theorem identifies the two.
For an e-process \((E_t)\) and \(\alpha {\gt}0\), \(P(E_t\ge 1/\alpha \text{ for some }t)\le \alpha \) for every \(P\in \mathcal{P}\), so that the test that rejects the null as soon as the e-process reaches \(1/\alpha \) has level \(\alpha \) (Section 1).
Markov’s inequality at the first time \(t\) with \(E_t\ge 1/\alpha \), which is a stopping time.
For every finite \(b\ge \eta _T\), the process \((E^{\star }_{t\wedge T}/b)_{t\ge 0}\) is an e-process with regret \(\log b\) at every \(t\le T\) (Section 5).
For a stopping time \(\tau \), \(\mathrm{E}_P(E^{\star }_{\tau \wedge T}/b)\le \eta _T/b\le 1\), and Lemma 15 gives the regret.
Let \(T\ge 0\). Then \(\Gamma _T\le \Gamma _{T+1}\). (i) For every \(t\ge 0\), the process with terms \(E_i=E^{\star }_t\mathbf{1}(i\ge t)\) is an e-process with \(R_t(E_t)=0\). (ii) For every \(\xi \in \Delta _T\), the process with terms \(\sum _{i=0}^{t\wedge T}\xi _iE^{\star }_i\) is an e-process with regret at most \(\log (1/\xi _t)\) at every \(t\le T\), so that \(\Gamma _T\le \log (T+1)\). (iii) If \(\eta _T{\lt}\infty \), then \((E^{\star }_{t\wedge T}/\eta _T)_{t\ge 0}\) is an e-process with regret \(\log \eta _T\) at every \(t\le T\), so that \(\Gamma _T\le \log \eta _T\).
As in Section S1.2. Part (ii) bounds \(E_\tau \le \sum _i\xi _iE^{\star }_i\) pointwise instead of writing the identity with \(\mathbf{1}(\tau \ge i)\). Part (iii) is Lemma 19 with \(b=\eta _T\), since \(\eta _T\ge 1\) by the stopping time \(\tau =0\).
If \(E^{\star }_0,\ldots ,E^{\star }_T\) have versions such that \((E^{\star }_{t\wedge T})_{t\ge 0}\) is an e-process, then \(\Gamma _T=0\).
By Lemma 15, the regret of this e-process is zero at every \(t\le T\), and \(\Gamma _T\ge 0\).
Let \(\mathcal{P}=\{ P\} \) satisfy the standing assumption. Then \(Q_t\ll P_t\), the numéraires \(E^{\star }_t=\mathrm{d}Q_t/\mathrm{d}P_t\) form a test martingale with respect to \((\mathcal{F}_t)\), and \(\Gamma _T=0\) for every \(T\) and every choice of numéraires (Section 1).
For a \(P_t\)-null set \(B\), \(1+\infty \mathbf{1}(X^t\in B)\) is an e-variable, which replaces \(1+m\mathbf{1}(X^t\in B)\) with \(m\to \infty \); the standing assumption then gives \(Q_t(B)=0\). The likelihood ratios form a martingale under \(P\), and \(\Gamma _T=0\) follows from Remark 1, with the stopped likelihood ratios as versions, instead of the cited Theorem 8 of Koolen and Grünwald (2022).
Two e-processes with finite regrets differing by \(o(t)\) have e-powers differing by \(o(t)\), so that \(t^{-1}\mathrm{E}_Q(\log E_t)\) and \(t^{-1}\mathrm{E}_Q(\log E'_t)\) have the same limit when one of them has a limit (Section 1).
By Lemma 15, \(R_t(E_t)-R_t(E'_t)=\mathrm{E}_Q(\log E'_t)-\mathrm{E}_Q(\log E_t)\).
For \(P\in \mathcal{P}\), \((\Lambda ^P_t)_{0\le t\le T}\) is the smallest supermartingale under \(P\) with respect to \((\mathcal{F}_t)\) that dominates \((E^{\star }_{t\wedge T})_{0\le t\le T}\), \(\Lambda ^P_0=\sup _\tau \mathrm{E}_P(E^{\star }_{\tau \wedge T})\), and \(\eta _T=\sup _{P\in \mathcal{P}}\Lambda ^P_0\) (Section 2).
The theorem of Peskir and Shiryaev. Since \(\mathcal{F}_0\) is trivial, \(\Lambda ^P_0\) is constant. Since \(\mathrm{E}_P(E^{\star }_t)\le 1\), \(E^{\star }_t{\lt}\infty \) \(P\)-almost surely, so that the real-valued process of the theorem has the same stopped expectations.
For every null and every choice of numéraires, \(\eta _T\) is nondecreasing in \(T\) (Example S1).
For a stopping time \(\tau \), \(\tau \wedge T\) is a stopping time with \((\tau \wedge T)\wedge (T+1)=\tau \wedge T\).
Let \((E_t)\) be a process with finite values, adapted to \((\mathcal{F}_t)\) and stopped at \(T\). Then \((E_t)\) is an e-process if and only if, for each \(P\in \mathcal{P}\), some supermartingale \((\Psi _t)_{0\le t\le T}\) under \(P\) with \(\Psi _t\ge E_t\) has initial value \(\Psi _0\le 1\) (Section S4, which states this for the Bernoulli tree).
If \((E_t)\) is an e-process, its Snell envelope dominates it, and its initial value is constant and at most one. Conversely, optional stopping for the supermartingale stopped at \(T\) gives \(\mathrm{E}_P(E_\tau )\le \Psi _0\le 1\).
Let \(P^{\star }_t=\int P_{\lambda ,t}\pi (\mathrm{d}\lambda )\) be a mixture of null distributions of \(X^t\), and let \(E\in \mathcal{E}_t\) satisfy \(E=\mathrm{d}Q_t/\mathrm{d}P^{\star }_t\). Then \(E\) is a numéraire at \(t\), and \(P^{\star }_t\) is the measure in (2) (Section S4, for the Bernoulli example, and the proof of Proposition 2).
For \(E'\in \mathcal{E}_t\), Fubini’s theorem gives \(\mathrm{E}_Q(E'/E)\le \mathrm{E}_{P^{\star }_t}(E')=\int \mathrm{E}_{P_\lambda }(E')\pi (\mathrm{d}\lambda )\le 1\).