Mean Deviation Bound for the Aggregate Fluctuation Covariance along a Mean-Field Trajectory Pair

Adopt the setting of the fluctuation processes of the controlled $N$-agent dynamics: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, and rate bound $B$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$, a mean-field trajectory pair $(S,A)$ for $\beta$ with horizon $T$, and the fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume that $\beta$ admits a twice continuously differentiable extension $(U,V,\bar{\beta})$ with derivative bound $K$, and that the control set $\mathcal{A}$ is convex. Let $\Theta$ be the aggregate fluctuation covariance of $\beta$, write $\mathbb{E}$ for the expectation and $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), set $z_s=(\mathfrak{s}_s,\mathfrak{a}_s)\in\mathbb{R}^{l+m}$ under the coordinate identification of the extension definition, and set

$$c_\Theta=2\,(l-1)\,\big(B+K\,\sqrt{l+m}\,\big).$$

Products with an infinite factor are read with the convention $0\cdot\infty=0$ (if $c_\Theta=0$, the rate bound forces $\beta$, and hence $\Theta$ and every left-hand side below, to vanish identically). Then, for all $\gamma,\delta\in\{1,\dots,l\}$:

(a) (Pointwise deviation bound.) At every $\omega\in\Omega$ and every $s\in[0,T]$,

$$\big|\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)-\Theta^{\gamma\delta}(S_s,A_s)\big|\ \le\ \frac{c_\Theta}{\sqrt{N}}\,|z_s| .$$

(b) (Mean deviation bound.) For every $s\in[0,T]$, with the expectation $\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]$ finite, and bounded and measurable as a function of $s$, by clause (a) of the weighted second-moment evolution lemma (applied with the constant matrix family $Z_t=0$, whose densities $\dot{z}^{\gamma\delta}\equiv0$ are continuous), and with the right-hand side valued in $[0,\infty]$,

$$\big|\mathbb{E}\big[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)\big]-\Theta^{\gamma\delta}(S_s,A_s)\big|\ \le\ \frac{c_\Theta}{\sqrt{N}}\,\Big(\mathbb{E}\big[|\mathfrak{s}_s|^2\big]+\mathbb{E}\big[|\mathfrak{a}_s|^2\big]\Big)^{1/2} .$$

(c) (Integrated deviation bound.) The function $s\mapsto\Theta^{\gamma\delta}(S_s,A_s)$ is continuous on $[0,T]$, relative to $[0,T]$, both $[0,T]$ and the codomain $\mathbb{R}$ carrying the metric of the real line — by continuity of compositions and continuity of sums and products along the continuous trajectory pair, the vector map $s\mapsto(S_s,A_s)$ being continuous into $\mathbb{R}^{l+m}$ in the Euclidean sense because $d((S_s,A_s),(S_t,A_t))\le\sum_{\gamma=1}^{l}|S^\gamma_s-S^\gamma_t|+\sum_{j=1}^{m}|A^j_s-A^j_t|$ by claim 1 of the componentwise estimates, each summand controlled by the componentwise continuity of the trajectory pair, and each $\beta(\sigma,\sigma',\cdot,\cdot)$, over ordered pairs $\sigma\neq\sigma'$, agreeing on the probability simplex-control product $\Delta^l\times\mathcal{A}$ with the continuous extension $\bar{\beta}(\sigma,\sigma',\cdot,\cdot)$, and the Euclidean and metric notions of continuity agreeing for real-valued maps by claim 1 of the continuity agreement lemma — and hence, by the extreme value theorem, attains a maximum and a minimum on $[0,T]$ and is in particular bounded there, so the function $s\mapsto\big|\mathbb{E}[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]-\Theta^{\gamma\delta}(S_s,A_s)\big|$ is measurable and bounded and its Lebesgue integral over $[0,T]$ is defined and finite. With

$$\mathcal{S}=\int_{[0,T]}\mathbb{E}\big[|\mathfrak{s}_t|^2\big]\,dt\qquad\text{and}\qquad\mathcal{A}_2=\int_{[0,T]}\mathbb{E}\big[|\mathfrak{a}_t|^2\big]\,dt$$

(the script $\mathcal{S}$ is distinct from the trajectory $S$, and $\mathcal{A}_2$, as in the a priori second-moment bound, from the control set $\mathcal{A}$), both well defined in $[0,\infty]$ with $\mathcal{S}\le4\,N\,T$ by clause (a) of the a priori second-moment bound and monotonicity of the integral, one has, in $[0,\infty]$,

$$\int_{[0,T]}\big|\mathbb{E}\big[\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)\big]-\Theta^{\gamma\delta}(S_s,A_s)\big|\,ds\ \le\ \frac{c_\Theta}{\sqrt{N}}\,\sqrt{T}\,\big(\mathcal{S}+\mathcal{A}_2\big)^{1/2} .$$