Proof of Anchored Pre-Stopping Envelope and Restricted Moment Bounds for the State Fluctuation Process

Throughout, "the envelope lemma" is the pre-stopping envelope lemma, "the anchored lemma" is the anchored good-set clocks lemma, "the tracking lemma" is the pathwise tracking lemma, and "the covariation lemma" is the stopped covariation lemma for the martingale part. All notation is that of the statement; $\mathfrak{b}_t$ and $\overline{\mathfrak{b}}=\varepsilon_Y+Q$ are the envelope and majorant of the envelope lemma.

Claim 1. Fix $\omega\in\Omega_0$ with $Y_{t_0}(\omega)<\varepsilon_1$ and $t\in[t_0,T]$, and put $u=\min(t,\sigma^*(\omega))$; then $u\in[t_0,T]$, since $\sigma^*$ is $[t_0,T]$-valued by claim 1 of the anchored lemma. By clause (vii)(a) of the existence and uniqueness theorem, $\Sigma_u(\omega)\in\Delta^l$ and $\Sigma_0(\omega)\in\Delta^l$. By the triangle inequality for the Euclidean norm,

$$|\Sigma_u(\omega)-S_u|\ \le\ \bigl|\Sigma_u(\omega)-S_u(\Sigma_0(\omega),\hat{\alpha}(\omega))\bigr|+\bigl|S_u(\Sigma_0(\omega),\hat{\alpha}(\omega))-S_u(x_0,\hat{\alpha}(\omega))\bigr|+Y_u(\omega),$$

since $S_u(x_0,\hat{\alpha}(\omega))=\Phi_u(\omega)$ and $Y_u(\omega)=|\Phi_u(\omega)-S^*_u|$ with $S^*=S$. The first summand is at most $|M_u(\omega)|+\Lambda_be^{\Lambda_bu}\mathcal{M}_u(\omega)\le|M_u(\omega)|+\Lambda_be^{\Lambda_bT}\mathcal{M}_u(\omega)$ by claim 2 of the tracking lemma (whose $S^\omega$ is the flow $S(\Sigma_0(\omega),\hat{\alpha}(\omega))$) and the fact that the exponential function is nondecreasing. The second summand is at most $e^{\Lambda_bT}|\Sigma_0(\omega)-x_0|=e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|$ by claim 4 of the flow stability lemma, applied with base pair $(x_0,\hat{\alpha}(\omega))$ and perturbed pair $(\Sigma_0(\omega),\hat{\alpha}(\omega))$ — the perturbed control equals the base control, so every $g^\gamma_r$ of claim 3 there vanishes and $G=0$ is admissible — together with $S_0=x_0$, which gives $|\Sigma_0(\omega)-x_0|=N^{-1/2}|\mathfrak{s}_0(\omega)|$. The third summand is at most $\varepsilon_1$: this is the stopped deviation bound of claim 2 of the anchored lemma, available since $Y_{t_0}(\omega)<\varepsilon_1$. Multiplying by $\sqrt{N}$ gives the first asserted inequality, $|\mathfrak{s}_u(\omega)|=\sqrt{N}|\Sigma_u(\omega)-S_u|$. For the second: by claim 1 of the envelope lemma, $\mathfrak{b}_s(\omega)\le\overline{\mathfrak{b}}(\omega)$ for every $s\in[0,T]$, and by the definitions of $\mathfrak{b}$ and $\overline{\mathfrak{b}}$ this reads

$$|M_s(\omega)|+\Lambda_be^{\Lambda_bT}\mathcal{M}_s(\omega)+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|=\mathfrak{b}_s(\omega)-\varepsilon_Y\le\overline{\mathfrak{b}}(\omega)-\varepsilon_Y=Q(\omega),$$

so the first bound is at most $\sqrt{N}(\varepsilon_1+Q(\omega))$. In particular, for $t<\sigma^*(\omega)$ one has $u=t$, so $\mathbf{1}_{\{t<\sigma^*\}}(\omega)|\mathfrak{s}_t(\omega)|\le\sqrt{N}(\varepsilon_1+Q(\omega))$; for $t\ge\sigma^*(\omega)$ the left side vanishes and the bound is trivial, $Q$ being nonnegative and $\varepsilon_1>Y_{t_0}(\omega)\ge0$ on the event considered.

Claim 2. By claim 2 of the covariation lemma, each $\mathbf{1}_{\Omega_0}\Sigma^\gamma$ is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, and $\sigma^*$ is a stopping time of that filtration (claim 1 of the anchored lemma), so by claim 4 of the stopping-time toolkit the sampled function $\omega\mapsto\mathbf{1}_{\Omega_0}(\omega)\Sigma^\gamma_{\min(t,\sigma^*(\omega))}(\omega)$ is a random variable; the function $\omega\mapsto S^\gamma_{\min(t,\sigma^*(\omega))}$ is a random variable as the composition of the measurable map $\omega\mapsto\min(t,\sigma^*(\omega))$ (for real $q$, $\{\min(t,\sigma^*)\le q\}$ is $\Omega$ or $\{\sigma^*\le q\}\in\mathcal{F}$) with the continuous $S^\gamma$, by measurability of continuous functions of measurable maps; hence $\mathbf{1}_{\Omega_0}\mathfrak{s}^\gamma_{\min(t,\sigma^*)}=\sqrt{N}(\mathbf{1}_{\Omega_0}\Sigma^\gamma_{\min(t,\sigma^*)}-\mathbf{1}_{\Omega_0}S^\gamma_{\min(t,\sigma^*)})$ and $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2$ are random variables by the same composition lemma. Likewise $\{s<\sigma^*\}\in\mathcal{G}_s\subseteq\mathcal{F}$ by claim 1 of the anchored lemma, and $\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2$ is a random variable ($\mathbf{1}_{\Omega_0}\mathfrak{s}^\gamma_s=\sqrt{N}(\mathbf{1}_{\Omega_0}\Sigma^\gamma_s-\mathbf{1}_{\Omega_0}S^\gamma_s)$ being a random variable, since $\mathbf{1}_{\Omega_0}\Sigma^\gamma_s$ is $\mathcal{F}^{\mathrm{sys}}_s$-measurable by progressive measurability and claim 1 of the progressive measurability toolkit, and $S^\gamma_s$ is a constant). Finally $Y_{t_0}$ is $\mathcal{G}_{t_0}$-measurable by claim 4 of the causality and adaptedness lemma, so $\{Y_{t_0}<\varepsilon_1\}\in\mathcal{G}_{t_0}$.

Claim 3. Let $s,t\in[t_0,T]$ and $D\in\mathcal{F}$ with $D\subseteq\{Y_{t_0}<\varepsilon_1\}$; $D\cap\Omega_0$ consists of points $\omega\in\Omega_0$ with $Y_{t_0}(\omega)<\varepsilon_1$. By claim 1, at every point of $D\cap\Omega_0$,

$$|\mathfrak{s}_{\min(t,\sigma^*)}|^2\le N(\varepsilon_1+Q)^2\qquad\text{and}\qquad\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2\le N(\varepsilon_1+Q)^2;$$

off $D\cap\Omega_0$ the integrands $\mathbf{1}_{D}\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2$ and $\mathbf{1}_{D}\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2$ vanish, and $\mathbf{1}_{D}N(\varepsilon_1+Q)^2$ is nonnegative. All integrands are random variables: claim 2; $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^4=(\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2)^2$ is a random variable as the composition of the continuous map $x\mapsto x^2$ with a random variable, by measurability of continuous functions of measurable maps; and $Q$ is a random variable by claim 2 of the envelope lemma, so $(\varepsilon_1+Q)^2$ and $(\varepsilon_1+Q)^4$ are random variables by the same composition lemma. All expectations below are finite: $Q$ is bounded ($0\le Q\le\sqrt{l}K_M+\Lambda_be^{\Lambda_bT}\sqrt{l}K_MT+2e^{\Lambda_bT}$, from $\overline{M}\le\sqrt{l}K_M$ by claim 2 of the envelope lemma, $0\le I\le\sqrt{l}\,K_MT$ everywhere by part (b) of the restricted-moments lemma, and $N^{-1/2}|\mathfrak{s}_0|\le2$ as recorded in the preamble of the envelope lemma), and the left-hand integrands are bounded by $4N$ and, in the fourth-moment display below, by $16N^2$: $|\Sigma_u-S_u|\le|\Sigma_u|+|S_u|\le2$, the squared Euclidean norm of a point of $\Delta^l$ being at most its coordinate sum $1$. Hence, by monotonicity of the expectation (linearity and monotonicity of the integral),

$$\mathbb{E}\bigl[\mathbf{1}_{D}\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^2\bigr]\le N\,\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^2\bigr],\qquad\mathbb{E}\bigl[\mathbf{1}_{D}\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\sigma^*\}}|\mathfrak{s}_s|^2\bigr]\le N\,\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^2\bigr].$$

Pointwise $(\varepsilon_1+Q)^2\le2\varepsilon_1^2+2Q^2$, so, by the Cauchy-Schwarz inequality for the mean-square norm applied to $\mathbf{1}_{D}$ and $Q^2$, together with $\mathbb{E}[\mathbf{1}_{D}^2]=P(D)$ and $\mathbb{E}[Q^4]\le c_Q\kappa_0N^{-2}$ (claim 2 of the envelope lemma),

$$\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^2\bigr]\le2\varepsilon_1^2P(D)+2\,\mathbb{E}\bigl[\mathbf{1}_{D}Q^2\bigr]\le2\varepsilon_1^2P(D)+2\,c_Q^{1/2}\kappa_0^{1/2}N^{-1}P(D)^{1/2};$$

multiplying by $N$ gives the second-moment bounds. For the fourth moments: applying $(a+b)^2\le2(a^2+b^2)$ twice gives $(\varepsilon_1+Q)^4\le8\varepsilon_1^4+8Q^4$ pointwise, and squaring the envelope of claim 1 twice gives $|\mathfrak{s}_{\min(t,\sigma^*)}|^4\le N^2(\varepsilon_1+Q)^4$ on $D\cap\Omega_0$, so

$$\mathbb{E}\bigl[\mathbf{1}_{D}\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\sigma^*)}|^4\bigr]\le N^2\,\mathbb{E}\bigl[\mathbf{1}_{D}(\varepsilon_1+Q)^4\bigr]\le N^2\bigl(8\varepsilon_1^4P(D)+8\,\mathbb{E}[Q^4]\bigr)\le8N^2\varepsilon_1^4P(D)+8\,c_Q\,\kappa_0 . \qquad\blacksquare$$