A Priori Second-Moment Bound for the State Fluctuation Process

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 associated fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume moreover that $\beta$ admits a twice continuously differentiable extension $(U,V,\bar{\beta})$ with derivative bound $K$, and that $\mathcal{A}$ is convex; the constant $\Lambda$ below is warranted by clauses (i) and (ii) of the regularity of the extended aggregate state drift. Let $b$ be the aggregate state drift of $\beta$, set $g_s=\sqrt{N}(b(\Sigma_s,\alpha_s)-b(S_s,A_s))$ with components $g^\gamma_s$, write $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ off $\Omega_0$, write $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), and set

$$\Lambda=l\,(B+K)\,\sqrt{l\,(l+m)}.$$

(a) (Well-definedness.) The maps $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{s}_t(\omega)|^2$ and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{a}_t(\omega)|^2$ are measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$ (by the joint measurability of the state and control), and so is $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)(g^\gamma_t(\omega))^2$ for each $\gamma$. Consequently, by the Tonelli theorem, the functions $t\mapsto\mathbb{E}[|\mathfrak{s}_t|^2]$, $t\mapsto\mathbb{E}[|\mathfrak{a}_t|^2]$, and $t\mapsto\mathbb{E}[(g^\gamma_t)^2]$ (expectations, which are unchanged by the $\mathbf{1}_{\Omega_0}$ modification because $\Omega_0$ has probability $1$) are measurable on $[0,T]$ as $[0,\infty]$-valued functions, with $\mathbb{E}[|\mathfrak{s}_t|^2]\le4N$ and $\mathbb{E}[(g^\gamma_t)^2]\le16N(l-1)^2B^2$ finite for every $t$, and the Lebesgue integral

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

is well defined with value in $[0,\infty]$ (the subscripted $\mathcal{A}_2$ is distinct from the control set $\mathcal{A}$).

(b) (Three-term estimate.) For every $t\in[0,T]$,

$$\mathbb{E}\big[|\mathfrak{s}_t|^2\big]\ \le\ 3\,\mathbb{E}\big[|\mathfrak{s}_0|^2\big]+6\,l\,(l-1)\,B\,T+3\,T\int_{[0,t]}\sum_{\gamma=1}^{l}\mathbb{E}\big[(g^\gamma_s)^2\big]\,ds .$$

(c) (A priori bound.) For every $t\in[0,T]$, with the exponential function,

$$\mathbb{E}\big[|\mathfrak{s}_t|^2\big]\ \le\ \Big(3\,\mathbb{E}\big[|\mathfrak{s}_0|^2\big]+6\,l\,(l-1)\,B\,T+3\,T\,\Lambda^2\,\mathcal{A}_2\Big)\,\exp\big(3\,T\,\Lambda^2\,t\big),$$

where the right-hand side is interpreted as $+\infty$ when $\mathcal{A}_2=+\infty$.