Pre-Stopping-Time Envelope and Restricted Moment Bounds for the State Fluctuation Process

Adopt the setting, notation and definitions of the extended good-set stopping-time lemma: the affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^m$, its transition-rate family $\beta$ with rate bound $B$ and state-Lipschitz constant $\Lambda_b$, the horizon $T>0$, the solution of the controlled $N$-agent dynamics with $N$ agents, regular event $\Omega_0$, empirical state measure $\Sigma$, observation filtration $(\mathcal{G}_t)_{t\in[0,T]}$ and system filtration $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, the realized control $\hat{\alpha}$, the point $x_0\in\Delta^l$, the realized mean-field flow $\Phi$, the map $S^*$ and deviation $Y$, the map $A$ and energy $\mathcal{E}$, the barriers $\varepsilon_Y$, $c_{\mathcal{E}}>0$, $\delta>0$, $\theta_{\mathrm{out}}>0$, the clipped-out time $\mathcal{O}$, the times $\tau_Y$, $\tau_{\mathcal{E}}$, $\tau_{\mathrm{out}}$, and the extended good-set time $\tau^*=\min(\tau_Y,\tau_{\mathcal{E}},\tau_{\mathrm{out}})$. Adopt additionally the setting, hypotheses and notation of the pathwise tracking lemma for the same affine-controlled transition-rate family, observation-rate family, horizon, driving system $(\Omega,\mathcal{F},P)$, policy, solution, and realized control $\hat{\alpha}$ — only claims 1 and 2 of that lemma are used here, so its population cost data may be taken arbitrary (for instance identically zero, which is convex in the control) and plays no role in the conclusions, and its hypothesis (LipC) is not assumed — including the martingale part $M=(M^1,\dots,M^l)$ of the martingale decomposition of the empirical state measure and the nondecreasing integrals $\mathcal{M}_t(\omega)=\int_{[0,t]}|M_s(\omega)|\,ds$ of claim 1 of the tracking lemma, defined for $\omega\in\Omega_0$; and the random variable $I$ of part (b) of the restricted-moments lemma for the martingale part, together with the constants $c_M$, $\kappa_T=BT+(BT)^2$ and $K_M=2+2(l-1)BT$ fixed in the preamble of that lemma, so that $I=\mathcal{M}_T$ on $\Omega_0$ and $\mathbb{E}[|M_t|^4]\le c_M\kappa_TN^{-2}$ and $\mathbb{E}[I^4]\le T^4c_M\kappa_TN^{-2}$.

Let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$ whose control is the map $A$ above (its components, being continuous, are measurable by claim 3 of the Borel toolkit for metric spaces) and whose initial state is $S_0=x_0$; take $S^*=S$. Let $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ be the state fluctuation process of the fluctuation processes of the controlled $N$-agent dynamics for this trajectory pair, and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$ its control fluctuation, $\alpha$ being the control of the solution. Assume $\varepsilon_Y>0$. Write $\mathbf{1}_{D}$ for the function equal to $1$ on a set $D$ and $0$ off it, $\mathbb{E}$ for the expectation, and set

$$\kappa_0=1+\mathbb{E}\bigl[|\mathfrak{s}_0|^4\bigr],$$

which is finite because $|\mathfrak{s}_0|\le2\sqrt{N}$ everywhere: every coordinate of a point of the probability simplex lies in $[0,1]$, so the squared Euclidean norm of a point is at most its coordinate sum $1$, and $|\Sigma_0-S_0|\le|\Sigma_0|+|S_0|\le2$.

For each $\gamma\in\{1,\dots,l\}$ let $\overline{M}^\gamma=\sup_{t\in D}\bigl(\mathbf{1}_{\Omega_0}|M^\gamma_t|\bigr)$ be the supremum random variable of the supremum lemma for bounded right-continuous processes applied to the process $(\mathbf{1}_{\Omega_0}M^\gamma_t)_{t\in[0,T]}$, with $D$ the set of dyadic partition points of $[0,T]$ of that lemma, and put $\overline{M}=\bigl(\sum_{\gamma=1}^{l}(\overline{M}^\gamma)^2\bigr)^{1/2}$. Define, for $\omega\in\Omega_0$ and $t\in[0,T]$ (the letter $\mathfrak{b}$ is used because $\beta$ names the rate family),

$$\mathfrak{b}_t(\omega)=\varepsilon_Y+|M_t(\omega)|+\Lambda_be^{\Lambda_bT}\mathcal{M}_t(\omega)+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|,$$

and, for every $\omega\in\Omega$, the envelope majorant

$$\overline{\mathfrak{b}}(\omega)=\varepsilon_Y+\overline{M}(\omega)+\Lambda_be^{\Lambda_bT}I(\omega)+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|\qquad\text{and}\qquad Q(\omega)=\overline{\mathfrak{b}}(\omega)-\varepsilon_Y\ \ge0 .$$

Then the following hold.

1. (Envelope.) For every $\omega\in\Omega_0$ and every $t\in[0,T]$,

$$\bigl|\mathfrak{s}_{\min(t,\tau^*(\omega))}(\omega)\bigr|\le\sqrt{N}\,\mathfrak{b}_{\min(t,\tau^*(\omega))}(\omega)\qquad\text{and}\qquad \mathfrak{b}_s(\omega)\le\overline{\mathfrak{b}}(\omega)\ \text{ for every }s\in[0,T];$$

in particular $\mathbf{1}_{\{t<\tau^*\}}(\omega)\,|\mathfrak{s}_t(\omega)|\le\sqrt{N}\,\overline{\mathfrak{b}}(\omega)$ for every $t\in[0,T]$ and $\omega\in\Omega_0$.

2. (Moments of the majorant.) $\overline{M}$, $I$ and $\overline{\mathfrak{b}}$ are random variables with $0\le\overline{M}\le\sqrt{l}\,K_M$ and

$$\mathbb{E}\bigl[\overline{M}^4\bigr]\le4\,l^2\,c_M\kappa_T\,N^{-2},\qquad \mathbb{E}\bigl[Q^4\bigr]\le c_Q\,\kappa_0\,N^{-2},\qquad c_Q=27\bigl(4l^2c_M\kappa_T+\Lambda_b^4e^{4\Lambda_bT}T^4c_M\kappa_T+e^{4\Lambda_bT}\bigr),$$

and consequently

$$\mathbb{E}\bigl[\overline{\mathfrak{b}}^2\bigr]\le2\varepsilon_Y^2+2\,c_Q^{1/2}\,\kappa_0^{1/2}\,N^{-1}\qquad\text{and}\qquad \mathbb{E}\bigl[\overline{\mathfrak{b}}^4\bigr]\le8\varepsilon_Y^4+8\,c_Q\,\kappa_0\,N^{-2}.$$

3. (Restricted second and fourth moments.) For all $s,t\in[0,T]$ the functions $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\tau^*)}|^2$ (with $\mathfrak{s}_{\min(t,\tau^*)}$ the componentwise sampled function, $\mathfrak{s}^\gamma_{\min(t,\tau^*)}(\omega)=\mathfrak{s}^\gamma_{\min(t,\tau^*(\omega))}(\omega)$) and $\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau^*\}}|\mathfrak{s}_s|^2$ are random variables, and for every event $D'\in\mathcal{F}$,

$$\mathbb{E}\bigl[\mathbf{1}_{D'}\mathbf{1}_{\Omega_0}\,|\mathfrak{s}_{\min(t,\tau^*)}|^2\bigr]\ \le\ N\,\mathbb{E}\bigl[\mathbf{1}_{D'}\overline{\mathfrak{b}}^2\bigr]\ \le\ 2N\varepsilon_Y^2\,P(D')+2\,c_Q^{1/2}\kappa_0^{1/2}\,P(D')^{1/2},$$

and the same bounds hold with $\mathbf{1}_{\Omega_0}|\mathfrak{s}_{\min(t,\tau^*)}|^2$ replaced by $\mathbf{1}_{\Omega_0}\mathbf{1}_{\{s<\tau^*\}}|\mathfrak{s}_s|^2$. Moreover

$$\mathbb{E}\bigl[\mathbf{1}_{\Omega_0}\,|\mathfrak{s}_{\min(t,\tau^*)}|^4\bigr]\ \le\ N^2\,\mathbb{E}\bigl[\overline{\mathfrak{b}}^4\bigr]\ \le\ 8N^2\varepsilon_Y^4+8\,c_Q\,\kappa_0 .$$

4. (Tail of the majorant.) For every real $\theta>0$,

$$P\bigl(\overline{\mathfrak{b}}\ge\varepsilon_Y+\theta\bigr)\ \le\ c_Q\,\kappa_0\,\theta^{-4}\,N^{-2}.$$

5. (Energy and the control fluctuation.) For every $\omega\in\Omega_0$ and every $t\in[0,T]$,

$$\mathcal{E}_t(\omega)=\frac{1}{N}\int_{[0,t]}|\mathfrak{a}_s(\omega)|^2\,ds .$$

6. (Unstopped envelope.) For every $\omega\in\Omega_0$ and every $t\in[0,T]$,

$$|\mathfrak{s}_t(\omega)|\ \le\ \sqrt{N}\bigl(Y_t(\omega)+Q(\omega)\bigr),$$

the deviation $Y_t$ appearing in place of the barrier $\varepsilon_Y$ and no stopping time appearing at all.