6 The lower bound
These are the results of the lower bound in Section S1.6 of the paper. The paper states them for \(\rho ^2\ge 1-s^2\), as in Proposition 3, and the proof of Proposition S5 uses them for every \(\rho {\gt}0\), with \(\Pi _{t_k}=\prod _{i=2}^{t_k}\ell (Y_i)\) in place of \(E^{\star }_{t_k}\). They are stated here in that generality: for every \(\rho {\gt}0\) and any numéraires of \(Q\).
Let \(s\in (0,1)\) and \(\rho {\gt}0\), fix integers \(r\ge 4\) and \(K\ge 1\), a horizon \(T\ge r^K\), and a constant \(c{\gt}0\). Let \(t_k=r^k\) for \(0\le k\le K\), let \(\xi \in \Delta _T\) be uniform on \(\{ t_0,\ldots ,t_K\} \), and let \(\gamma ^2=\rho ^2+s^2\). Let \(\theta \sim \mathrm{N}(0,\gamma ^2)\), let \(X\sim P_\theta \) given \(\theta \), and let \(Z_t=t^{1/2}(\bar X_t-\theta )\). Let \(\tau '\) be the first \(t_k\) with \(c\le |Z_{t_k}|\le c+1\), or \(\infty \) if there is none. With probability \(1/2\), let \(\tau =\tau '\), and otherwise let \(\tau \sim \xi \) be independent of \(X\); let \(\bar P\) be the distribution of \((\tau ,X^\tau )\). For \(j{\lt}k\) and \(t=t_k\), let \(W_{jk}(y)=t_j^{1/2}\sum _{i=t_j+1}^ty_i/\{ i(i-1)\} ^{1/2}\), let \(\mathcal{Z}_k(y)=\{ z:c\le |z|\le c+1\} \smallsetminus \bigcup _{j{\lt}k}\{ z:c\le |(t_j/t)^{1/2}z-W_{jk}(y)|\le c+1\} \), and let \(p_k(y)=\int _{\mathcal{Z}_k(y)}\varphi _1(z)\mathrm{d}z\). Let \(c'=c-(c+1)r^{-1/2}\), \(A_k=\{ y:\max _{j{\lt}k}|W_{jk}(y)|{\lt}c'\} \), \(\beta '=\frac12\log \{ \rho ^2/(\gamma ^2+1)\} -\log 2\), and \(\beta =\log (K+1)+\log \varphi _1(c+1)+\log (\rho /\gamma )\).
For \(0\le k\le K\) and \(t=t_k\),
For every \(\theta \) and \(0\le k\le K\), with \(t=t_k\), \(\tau '=t\) exactly when \(Z_t\in \mathcal{Z}_k(Y^t)\). Since \(Z_t\sim \mathrm{N}(0,1)\) is independent of \(Y^t\) under \(P_\theta \), \(p_k(Y^t)\) is the conditional probability that \(\tau '=t\), given \(Y^t\). If \(c'{\gt}0\), then, on the event \(\{ Y^t\in A_k\} \), the rule \(\tau '\) stops at \(t\) whenever \(c\le |Z_t|\le c+1\) (Section S1.6, after Lemma S5).
By Lemma S2(ii), \(Z_{t_j}=(t_j/t)^{1/2}Z_t-W_{jk}(Y^t)\) for \(j{\lt}k\). Under \(P_\theta \), \((Z_t,Y^t)\) has the distribution \(\mathrm{N}(0,1)\otimes \mathrm{N}(0,1)^{\otimes (t-1)}\) by Lemma S2(iii), and the conditional expectation is characterized by its set integrals. On \(A_k\), \(|(t_j/t)^{1/2}z-W_{jk}(y)|{\lt}(c+1)r^{-1/2}+c'=c\) for \(|z|\le c+1\), so that \(\mathcal{Z}_k(y)=\{ z:c\le |z|\le c+1\} \).
For every \(\rho {\gt}0\) and any numéraires of \(Q\), the distribution \(\bar P\) belongs to \(\mathcal{B}\), and
which is eq. (S8).
As in Section S1.6: \(\mathrm{d}Q^\xi /\mathrm{d}\bar P=\xi _t\Pi _t/J_k\) on the pairs with time \(t=t_k\), so that \(\mathrm{KL}(Q^\xi \| \bar P)=\sum _k\xi _{t_k}\mathrm{E}_Q(\log \Pi _{t_k})-\sum _k\xi _{t_k}\mathrm{E}_Q\{ \log (J_k/\xi _{t_k})\} \), and Lemma 1 with \(\xi _{t_k}=1/(K+1)\) gives (S8), since \(\mathrm{E}_Q(\log E^{\star }_t)\ge \mathrm{E}_Q(\log \Pi _t)\) by log-optimality and \(\Pi _t\in \mathcal{E}_t\). The densities are taken with respect to \(\mathrm{d}\bar x_t\, \mathrm{N}(0,1)^{\otimes (t-1)}(\mathrm{d}y)\), through the distributions of \((\bar X_t,Y^t)\) under \(P_\theta \) and \(Q\), instead of Lebesgue densities on \(\mathbb {R}^t\); the Gaussian identity \(\int \varphi _{1/t}(a-\theta )\varphi _{\gamma ^2}(\theta )\mathrm{d}\theta =\varphi _{\gamma ^2+1/t}(a)\) is proved by completing the square. The steps that the paper leaves implicit are proved: the probability \(1/(2K+2)\) of each uniform horizon, the measurability of \(\tau '\), the \(Q\)-integrability of \(\log \Pi _t\) and of \(\log \{ (K+1)J_k\} \), and the nonnegativity of the sum that gives \(\mathrm{KL}(Q^\xi \| \bar P)\).
Let \(\rho {\gt}0\), \(0\le k\le K\), and \(t=t_k\). (i) For every \(x\in \mathbb {R}^t\), \(J_k(x){\gt}0\) and \(\log \{ (K+1)J_k(x)\} \ge \beta '\). (ii) For every \(x\in \mathbb {R}^t\), \(J_k(x)\ge \{ \rho /(2\gamma )\} \exp [-\{ 2(c+1)t^{-1/2}|\bar x_t|+(c+1)^2t^{-1}\} /(2\gamma ^2)]p_k(y)\). (iii) If \(c'{\gt}0\), then \(p_k(y)\ge 2\varphi _1(c+1)\) for every \(y\in A_k\).
As in Section S1.6. (i) For \(0{\lt}v_1\le v_2\), \(\varphi _{v_2}/\varphi _{v_1}\ge (v_1/v_2)^{1/2}\), applied to the second term of \(J_k\). (ii) A bound on \(\varphi _{\gamma ^2}(\bar x_t-t^{-1/2}z)/\varphi _{\rho ^2+s^2/t}(\bar x_t)\) for \(|z|\le c+1\). (iii) On \(A_k\), \(\mathcal{Z}_k(y)=\{ z:c\le |z|\le c+1\} \), and \(p_k(y)\ge \varphi _1(c+1)\) times the Lebesgue measure \(2\) of this set, instead of \(2\int _c^{c+1}\varphi _1\).
For every \(\rho {\gt}0\) and any numéraires of \(Q\), if \(c'{\gt}0\) and \(\beta \ge \beta '\), then
which is eq. (S9).
As in Section S1.6, from (S8): the two bounds of Lemma S5 are combined through \(\mathbf{1}(Y^t\notin A_k)\), and \(Q(Y^t\notin A_k)\le 2K\exp \{ -c'^2/(2s^2)\} \) by the Gaussian tail bound \(\mathrm{pr}(|N|\ge x)\le \exp \{ -x^2/(2v)\} \) for \(N\sim \mathrm{N}(0,v)\), proved from \(\varphi _1(u+x)\le e^{-x^2/2}\varphi _1(u)\). The variance of \(W_{jk}\) under \(Q\) is bounded by \(s^2\) instead of the exact \(s^2(1-t_j/t)\), and \(\mathrm{E}_Q|\bar X_t|\le (\rho ^2+s^2/t)^{1/2}\) follows from \(|a|\le (a^2+v)/(2v^{1/2})\) instead of Jensen’s inequality; both give the same bound. The statement "(S8) implies (S9)" holds for any value in place of \(\Gamma _T\).