Proof of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound

Preliminaries. Fix a natural number $N\ge1$ and an admissible parameter vector $\pi$, and let $K$, the grid, the levels, the clocks, the good sets, the leave events, the block energy increments, $\Lambda_{\star}$, $\mathcal{Z}$, $\mathcal{P}$ and $\Upsilon_{\mathrm{lev}}$ be the objects formed from $\pi$ for the $N$-th solution. By the definition of admissibility every parameter-dependent hypothesis of the ledger lemma holds for that solution, and its remaining standing hypotheses hold by assumption; so the ledger lemma, and with it the block cascade lemma whose setting it adopts, applies to the $N$-th solution for every $N\ge1$. Four facts are used repeatedly.

(P1) By claim 5 of the progressive measurability lemma for the realized control, applied to the map $A$ of the setting, each of whose components is measurable with respect to the trace Borel $\sigma$-algebra $\mathcal{B}_{[0,T]}$ on $[0,T]$ and the Borel $\sigma$-algebra of the real line: for every $t\in[0,T]$ the function $\mathcal{E}_{t}$ is $\mathcal{G}_{t}$-measurable with values in $[0,4R^{2}T]$, and $0\le\mathcal{E}_{t}(\omega)-\mathcal{E}_{t_{0}}(\omega)\le4R^{2}(t-t_{0})$ for all $0\le t_{0}\le t\le T$ and every $\omega\in\Omega$. In particular every path $t\mapsto\mathcal{E}_{t}(\omega)$ is nondecreasing, and $\mathcal{E}_{0}=0$ because the integral over $[0,0]$ is $0$.

(P2) By claim 1 of the block cascade lemma, $K\ge1$, $t_{0}=0$, $t_{K}=T$ and $t_{k}<t_{k+1}$ for every $k\in\{0,\dots,K-1\}$.

(P3) By claim 2 of the block cascade lemma each clock $\sigma^{(k)}$ takes its values in $[t_{k},T]$, each $G_{k}$ is an event, $G_{0}\supseteq G_{1}\supseteq\dots\supseteq G_{K}$, and the sets $D_{0},\dots,D_{K-1},G_{K}$ are pairwise disjoint with union $\Omega_{0}$; moreover $G_{k+1}=G_{k}\cap\{\sigma^{(k)}\ge t_{k+1}\}$ by the definition of the good sets there.

(P4) By (P1), (P2) and (P3), $t_{k}\le\min(\sigma^{(k)}(\omega),t_{k+1})\le T$ for every $k$ and every $\omega$, so $\Delta_{k}\mathcal{E}\ge0$; each $\Delta_{k}\mathcal{E}$ is a random variable with values in $[0,4R^{2}h_{k}]$, as recorded in the block cascade lemma. Hence every expectation appearing in the definition of $\mathcal{Z}$ is that of a bounded nonnegative random variable, and $\mathcal{Z}$ is a finite nonnegative real number.

Proof of claim 1. Let $\omega\in G_{K}$ and $k\in\{0,\dots,K-1\}$. By (P3) we have $G_{K}\subseteq G_{k+1}\subseteq\{\sigma^{(k)}\ge t_{k+1}\}$, so $\sigma^{(k)}(\omega)\ge t_{k+1}$, hence $\min(\sigma^{(k)}(\omega),t_{k+1})=t_{k+1}$ and

Summing over $k\in\{0,\dots,K-1\}$, the right-hand sides telescope, so by (P2) and $\mathcal{E}_{0}=0$,

Multiplying by the indicator of $G_{K}$, and noting that both sides vanish off $G_{K}$, we get the identity $\mathbf{1}_{G_{K}}\mathcal{E}_{T}=\sum_{k=0}^{K-1}\mathbf{1}_{G_{K}}\Delta_{k}\mathcal{E}$ of functions on $\Omega$. Since $G_{K}\subseteq G_{k}$ by (P3) and $\Delta_{k}\mathcal{E}\ge0$ by (P4), we have $\mathbf{1}_{G_{K}}\Delta_{k}\mathcal{E}\le\mathbf{1}_{G_{k}}\Delta_{k}\mathcal{E}$ at every point of $\Omega$. All the functions involved are bounded random variables, hence have finite expectations, so by the additivity and monotonicity of the expectation

the last equality being the definition of $\mathcal{Z}$.

For the second assertion, $P(\Omega_{0})=1$ by the definition of a solution of the controlled $N$-agent dynamics, and by (P3) the events $D_{0},\dots,D_{K-1},G_{K}$ are pairwise disjoint with union $\Omega_{0}$; so by the additivity of the probability, $1=P(\Omega_{0})=\mathcal{P}+P(G_{K})$ and therefore $P(\Omega\setminus G_{K})=1-P(G_{K})=\mathcal{P}$. The inequality $\mathcal{P}\le\Lambda_{\star}\mathcal{Z}\Upsilon_{\mathrm{lev}}N^{-1}$ is the first assertion of clause (d) of claim 1 of the ledger lemma. Finiteness of the three quantities holds by (P4) and because probabilities are finite.

Proof of claim 2. Fix $N\ge1$, $\omega\in\Omega$ and $t\in[0,T]$, and define $f:[0,T]\to\mathbb{R}$ by $f(s)=|\hat{\alpha}(s,\omega)-A_{s}|$.

The function $f$ is measurable and bounded. By claim 3 of the realized-control lemma the path $s\mapsto\hat{\alpha}(s,\omega)$ takes all its values in $\mathcal{A}$ and is square-integrable; in particular each of its components is $\mathcal{B}_{[0,T]}$-measurable, by the definition of the Lebesgue space of square-integrable vector-valued maps. Each component of $A$ is $\mathcal{B}_{[0,T]}$-measurable, as recorded in (P1). Hence by claim 2 of the arithmetic of measurable real functions each component of $s\mapsto\hat{\alpha}(s,\omega)-A_{s}$ is $\mathcal{B}_{[0,T]}$-measurable, and since the Euclidean norm is continuous on $\mathbb{R}^{m}$, hence sequentially continuous there, the composition lemma for sequentially continuous functions of measurable Euclidean maps shows that $f$ is $\mathcal{B}_{[0,T]}$-measurable. Both $\hat{\alpha}(s,\omega)$ and $A_{s}$ lie in $\mathcal{A}$ and $|a|\le R$ for every $a\in\mathcal{A}$, so $0\le f\le2R$ by the triangle inequality for the Euclidean norm; and $f^{2}$, which is $\mathcal{B}_{[0,T]}$-measurable by claim 3 of the arithmetic lemma, satisfies $\int_{[0,t]}f^{2}\,ds=\mathcal{E}_{t}(\omega)$, a finite number by (P1).

The first bound. If $t=0$ both sides vanish, since the integral over $[0,0]$ is $0$ and $\mathcal{E}_{T}(\omega)\ge0$. Let $t>0$ and apply claim 4 of the integral toolkit on the compact interval $[0,t]$ to the restriction of $f$ and to the constant function $1$: these are $\mathcal{B}_{[0,t]}$-measurable with $\int_{[0,t]}f^{2}\,ds<\infty$ and $\int_{[0,t]}1\,ds=t<\infty$, so $f$ is integrable on $[0,t]$ and

the last step by $t\le T$ and the monotonicity of $u\mapsto\mathcal{E}_{u}(\omega)$ from (P1). Taking nonnegative square roots, and using that the nonnegative square root is nondecreasing, gives $\int_{[0,t]}f\,ds\le\sqrt{T}\,\mathcal{E}_{T}(\omega)^{1/2}$.

The second bound. Apply claim 4 of the extended good-set stopping-time lemma in the instance of its setting determined by the present data with $S^{*}=S$, the thresholds $\delta$, $\theta_{\mathrm{out}}$, $\varepsilon_{Y}$ and $c_{\mathcal{E}}$ of that setting being given the value $1$, which is permitted because none of $\hat{\alpha}$, $\Phi$, $Y$, $\mathcal{E}$, $C_{S}$ and none of the conclusions of that claim involves them. The standing hypothesis on $S^{*}$ required by that claim holds here: $(S,A,P)$ is a stationary mean-field triple with $S_{0}=x_{0}$, so $S$ takes its values in $\Delta^{l}$ and $S^{\gamma}_{t}=x^{\gamma}_{0}+\int_{[0,t]}b^{\gamma}(S_{s},A_{s})\,ds$ for every $t\in[0,T]$ and every $\gamma\in\{1,\dots,l\}$. That claim gives $Y_{t}(\omega)\le C_{S}\,\mathcal{E}_{t}(\omega)^{1/2}$; since $\mathcal{E}_{t}(\omega)\le\mathcal{E}_{T}(\omega)$ by (P1) and the nonnegative square root is nondecreasing, and since $Y_{t}(\omega)=|\Phi_{t}(\omega)-S_{t}|$, we get $|\Phi_{t}(\omega)-S_{t}|\le C_{S}\,\mathcal{E}_{T}(\omega)^{1/2}$.

Neither bound refers to the parameter vector: $\hat{\alpha}$, $A$, $\mathcal{E}$, $\Phi$, $S$ and $C_{S}$ are all defined without it.

Proof of claim 3. Let $\mathsf{A}$ be the collection of those parameter vectors that are admissible and satisfy the absorption condition of clause (a) of the lower bound theorem. Both requirements involve only the parameter vector and constants of the common data: admissibility does so by the remark to that effect in the lower bound theorem, and the absorption condition $2C^{\vee}_{tg}\Lambda_{\star}+2\epsilon C_{S}^{2}\le c_{\star}/4$ because $\epsilon$ is one of the components of the parameter vector, $C_{S}$ is determined by the common data and $\Lambda_{\star}$ by the parameter vector and the common data, and $C^{\vee}_{tg}$ and $c_{\star}$ are among the constants that the lower bound theorem lists as determined by the common data and the same for every $N$. Hence $\mathsf{A}$ is determined by the common data alone, and in particular depends neither on $N$ nor on the individual solutions. It is nonempty, because clause (b) applied with $\varepsilon'=1$ furnishes a member of it. Choose $\pi_{\bullet}\in\mathsf{A}$, put $\mathcal{Z}^{\sharp}=\mathcal{Z}^{\sharp}(\pi_{\bullet})$, and let $N_{1}$ be the natural number that clause (a), applied to $\pi_{\bullet}$, furnishes; then for every $N\ge N_{1}$ the tracked energy of the $N$-th solution formed from $\pi_{\bullet}$ satisfies $\mathcal{Z}\le\mathcal{Z}^{\sharp}$. Now $\pi_{\bullet}$ has been chosen from a collection determined by the common data; the real number $\mathcal{Z}^{\sharp}(\pi)$ is defined from $\pi$, from the bounds $\mathcal{J}^{\sharp}$ and $\kappa^{\sharp}$ and from constants of the common data; and clause (a) provides $N_{1}$ depending on $\pi_{\bullet}$ and the common data but not on the solutions. So $\pi_{\bullet}$, $\mathcal{Z}^{\sharp}$ and $N_{1}$ are determined as asserted. Finally, taking $N=N_{1}$ gives $\mathcal{Z}^{\sharp}\ge\mathcal{Z}\ge0$ by (P4), so $\mathcal{Z}^{\sharp}\ge0$.

Proof of claim 4. Such a number $N_{\flat}$ exists: by clause 1 of the Archimedean property there is a natural number $N'$ with $N'\ge C_{\mathrm{esc}}/\varepsilon_{\flat}$, and the larger of $N'$ and $N_{1}$ is then a natural number $N_{\flat}$ with $N_{\flat}\ge N_{1}$ and $\varepsilon_{\flat}N_{\flat}\ge\varepsilon_{\flat}N'\ge C_{\mathrm{esc}}$. Fix $N\ge N_{\flat}$ and write $c=C_{\mathrm{ctl}}/(\varepsilon_{\flat}N)$, a positive real number because $C_{\mathrm{ctl}}\ge2>0$ by claim 3, $\varepsilon_{\flat}>0$ and $N\ge1$.

$\Omega^{\flat}_{N}$ is an event of $\mathcal{G}_{T}$. By claim 1 of the cascade filtering lemma the good set $G_{K}$ belongs to $\mathcal{G}_{t_{K}}$, and $t_{K}=T$ by (P2). By (P1) the function $\mathcal{E}_{T}$ is $\mathcal{G}_{T}$-measurable, so $\{\mathcal{E}_{T}\le c\}\in\mathcal{G}_{T}$. A $\sigma$-algebra is closed under intersections and $\mathcal{G}_{T}\subseteq\mathcal{F}$, so $\Omega^{\flat}_{N}\in\mathcal{G}_{T}$ and $\Omega^{\flat}_{N}$ is an event.

The probability bound. Since $\Omega^{\flat}_{N}$ is the intersection of $G_{K}$ with $\{\mathcal{E}_{T}\le c\}$,

and it suffices to bound the probability of each of the two sets on the right by $\varepsilon_{\flat}/2$.

For the first, claim 1 and claim 3 give $P(\Omega\setminus G_{K})\le\Lambda_{\star}\Upsilon_{\mathrm{lev}}\mathcal{Z}N^{-1}\le\Lambda_{\star}\Upsilon_{\mathrm{lev}}\mathcal{Z}^{\sharp}N^{-1}=C_{\mathrm{esc}}/(2N)$, the middle step because $\Lambda_{\star}>0$ and $\Upsilon_{\mathrm{lev}}>0$ and $\mathcal{Z}\le\mathcal{Z}^{\sharp}$ for $N\ge N_{1}$; and $\varepsilon_{\flat}N\ge\varepsilon_{\flat}N_{\flat}\ge C_{\mathrm{esc}}$ gives $C_{\mathrm{esc}}/(2N)\le\varepsilon_{\flat}/2$.

For the second, put $X=\mathbf{1}_{G_{K}}\mathcal{E}_{T}$, a nonnegative bounded random variable with $\mathbb{E}[X]\le\mathcal{Z}/N\le\mathcal{Z}^{\sharp}/N$ by claims 1 and 3. If $\omega\in G_{K}$ and $\mathcal{E}_{T}(\omega)>c$ then $X(\omega)=\mathcal{E}_{T}(\omega)>c$, so $G_{K}\cap\{\mathcal{E}_{T}>c\}\subseteq\{X\ge c\}$. Hence, by Markov's inequality applied to $X$ and the positive real $c$, and by the monotonicity of the probability,

the last step because $\mathcal{Z}^{\sharp}\ge0$ and hence $\mathcal{Z}^{\sharp}\le\mathcal{Z}^{\sharp}+1$. Adding the two bounds and using the subadditivity of the probability gives $P(\Omega\setminus\Omega^{\flat}_{N})\le\varepsilon_{\flat}$.

The three pathwise bounds. Let $\omega\in\Omega^{\flat}_{N}$ and $t\in[0,T]$. By (P1) and the definition of $\Omega^{\flat}_{N}$,

which is the first bound. Since the nonnegative square root is nondecreasing, $\mathcal{E}_{T}(\omega)^{1/2}\le c^{1/2}$, so claim 2 gives

which are the second and third bounds. This completes the proof.