Proof of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift

Throughout fix $\upsilon\in\{1,\dots,\tilde{l}\}$ and recall from the extension definition its clauses 1-4. On $\Delta^l$ we use, from the probability simplex: $\Sigma^\sigma\ge0$ for all $\sigma$ and $\sum_{\sigma=1}^{l}\Sigma^\sigma=1$; and from clause 1 together with the observation-rate family bounds: $0\le\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)\le\tilde{B}$ for $\Sigma\in\Delta^l$ and all admissible pairs. We write $\mathbf{1}_{\{\cdot\}}$ for the indicator equal to $1$ when the subscripted condition holds and $0$ otherwise. The argument is the same as, but logically independent of, the proof of the regularity and derivative bounds of the extended aggregate state drift, with the control coordinates absent; in part (iii) the segments used here lie in $\Delta^l$, so no convexity of $\tilde{U}$ is needed.

(i). We first record a slice principle. Fix a point $a\in\tilde{U}$; since $\tilde{U}$ is open, there is a real $\rho>0$ such that every point of $\mathbb{R}^{l}$ at Euclidean distance less than $\rho$ from $a$ lies in $\tilde{U}$. For $\gamma\in\{1,\dots,l\}$ and $\tau\in\mathbb{R}$ write $a+\tau e_\gamma$ for the point obtained by adding $\tau$ to the $\gamma$th coordinate of $a$; its distance from $a$ is $|\tau|$ by claims 1 and 2 of the norm properties, so $a+\tau e_\gamma\in\tilde{U}$ for $\tau$ in the interval $J=(-\rho,\rho)$. For a function $g$ on $\tilde{U}$ and the slice $G:J\to\mathbb{R}$, $G(\tau)=g(a+\tau e_\gamma)$, the partial derivative of $g$ with respect to the $\gamma$th variable exists at $a$ with value $L$ if and only if $G$ is differentiable at $0$ with $G'(0)=L$: both are the same $\varepsilon$-$\delta$ condition on the quotients $(g(a+\tau e_\gamma)-g(a))/\tau$, the membership requirement in the partial-derivative definition holding for $0<|\tau|<\rho$ by the choice of $\rho$.

For $\sigma\in\{1,\dots,l\}$ let $\pi_\sigma:\tilde{U}\to\mathbb{R}$ be the coordinate function $\pi_\sigma(\Sigma)=\Sigma^\sigma$. Its slice difference quotients of the preceding paragraph are constantly $\mathbf{1}_{\{\gamma=\sigma\}}$, so $\partial_\gamma\pi_\sigma$ exists at every point and equals the constant $\mathbf{1}_{\{\gamma=\sigma\}}$; moreover $\pi_\sigma$ is continuous at every point, since $|\pi_\sigma(\Sigma)-\pi_\sigma(\Sigma')|\le d(\Sigma,\Sigma')$ by claims 2 and 4 of the norm properties, as are the constant functions; hence $\pi_\sigma$ and the constants are of class $C^1$ on $\tilde{U}$ (clauses 1 and 3 of that definition). By the definition of the extended aggregate observation drift,

Fix $\gamma$ and a point $a\in\tilde{U}$, with $J=(-\rho,\rho)$ as in the slice principle. Each function $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ is of class $C^2$, hence of class $C^1$, on $\tilde{U}$ by clause 2 of the extension definition and clause 2 of the $C^k$ definition; so by claim 2 of the segment-derivative lemma, applied with $W=\tilde{U}$, $x=a$ and $h=e_\gamma$, its slice on $J$ is differentiable at every interior point $\tau_0$ of $J$ with derivative $\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,a+\tau_0 e_\gamma)$, and the same holds for $\pi_\sigma$ with derivative $\mathbf{1}_{\{\gamma=\sigma\}}$. The one-dimensional sum, constant-multiple and product rules then give that the slice of $\bar{\tilde{b}}^\upsilon$ is differentiable at $0$; by the slice principle, $\partial_\gamma\bar{\tilde{b}}^\upsilon$ exists at $a$ and, writing $\Sigma=a$, equals

the first displayed formula of the statement. Each factor appearing in that formula is continuous at every point of $\tilde{U}$ in the sense of Euclidean continuity: $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ and its first partial derivatives by clauses 1 and 2 of the $C^k$ definition (the family being of class $C^2$), and the coordinate functions and constants as recorded above. By claim 1 of the continuity agreement lemma each of these functions is continuous on $\tilde{U}$ relative to $\tilde{U}$ as a map into the real line, so their finite sums and products are continuous on $\tilde{U}$ by claim 5 of continuity of sums and products, and claim 1 of the agreement lemma translates this back into Euclidean continuity at every point. Hence $\partial_\gamma\bar{\tilde{b}}^\upsilon$ is continuous for every $\gamma$, and $\bar{\tilde{b}}^\upsilon$ itself is continuous by the same two-way translation applied to its defining formula; so $\bar{\tilde{b}}^\upsilon$ is of class $C^1$ on $\tilde{U}$ (clause 1 of the $C^k$ definition). Applying the same slice argument to the first displayed formula --- a finite sum of products of coordinate functions, the functions $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$, and their first partials, all of class $C^1$ on $\tilde{U}$ (the first partials by clause 2 of the extension definition with clause 2 of the $C^k$ definition) --- gives that $\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon$ exists at every point and equals

which is the second displayed formula after evaluating the indicator sum, and this expression is continuous by the same reasoning, the second partials $\partial_\delta\partial_\gamma\bar{\tilde{\beta}}$ being continuous by clauses 1 and 2 of the $C^k$ definition; hence each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ is of class $C^1$ on $\tilde{U}$, and $\bar{\tilde{b}}^\upsilon$ is of class $C^2$ there by clause 2 of the $C^k$ definition. Finally, for $\Sigma\in\Delta^l$, clause 1 allows replacing every $\bar{\tilde{\beta}}$ by $\tilde{\beta}$ in the defining formula of $\bar{\tilde{b}}^\upsilon(\Sigma)$, which then coincides with the defining formula of the aggregate observation drift of $\tilde{\beta}$ at $\Sigma$; hence $\bar{\tilde{b}}$ agrees with it on $\Delta^l$.

(ii). Let $\Sigma\in\Delta^l$ and $\gamma\in\{1,\dots,l\}$. In the first displayed formula, $|\bar{\tilde{\beta}}(\gamma,\upsilon,\Sigma)|\le\tilde{B}$ as recorded above, and $\sum_{\sigma}\Sigma^\sigma|\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)|\le\tilde{K}\sum_{\sigma}\Sigma^\sigma=\tilde{K}$ by clause 3; hence $|\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le\tilde{B}+\tilde{K}$. For the Lipschitz estimate let $\Sigma,\Sigma'\in\Delta^l$. The segment from $\Sigma$ to $\Sigma'$ stays in $\Delta^l$: a convex combination of two points of the simplex has nonnegative entries summing to $1$, hence lies in the simplex. The segment therefore lies in the open set $\tilde{U}$, and part (i) of the Taylor expansion lemma, applied to the $C^1$ function $\bar{\tilde{b}}^\upsilon$ with $n=l$ and $M_1=\tilde{B}+\tilde{K}$ (the first-order bound just proved holding at every point of the segment, these lying in $\Delta^l$), gives $|\bar{\tilde{b}}^\upsilon(\Sigma)-\bar{\tilde{b}}^\upsilon(\Sigma')|\le\sqrt{l}\,(\tilde{B}+\tilde{K})\,d(\Sigma,\Sigma')$.

(iii). For $\Sigma\in\Delta^l$, estimating the three groups of the second displayed formula with clause 3 and $\sum_{\sigma}\Sigma^\sigma=1$: the two first-derivative terms contribute at most $\tilde{K}$ each, and the sum $\sum_\sigma\Sigma^\sigma\,\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)$ contributes at most $\tilde{K}$ in total; hence $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le3\tilde{K}$.

For the uniform continuity claim, let $\varepsilon>0$ and set $C^*=2\sqrt{l}\,\tilde{K}+l\,\tilde{K}$. By clause 4 there is $\delta_1>0$ such that $|\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)-\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma')|\le\varepsilon/2$ for all admissible indices whenever $d(\Sigma,\Sigma')\le\delta_1$. Set $\delta^\circ=\delta_1$ if $C^*=0$ and $\delta^\circ=\min(\delta_1,\varepsilon/(2C^*))$ otherwise. Let $\Sigma,\Sigma'\in\Delta^l$ with $d(\Sigma,\Sigma')\le\delta^\circ$, and take the difference of the second displayed formula at $\Sigma$ and at $\Sigma'$ term by term. For the two first-derivative terms: each of the two functions $\partial_\delta\bar{\tilde{\beta}}(\gamma,\upsilon,\cdot)$ and $\partial_\gamma\bar{\tilde{\beta}}(\delta,\upsilon,\cdot)$ is of class $C^1$ on $\tilde{U}$ (clause 2 of the extension definition with clause 2 of the $C^k$ definition) with partial derivatives bounded by $\tilde{K}$ on $\tilde{U}$ (clause 3), so part (i) of the Taylor expansion lemma along the segment from $\Sigma$ to $\Sigma'$ (which lies in the simplex, as above) bounds each difference by $\sqrt{l}\,\tilde{K}\,d(\Sigma,\Sigma')$. For the product terms,

and summing over $\sigma$, using $|\Sigma^\sigma-\Sigma'^\sigma|\le d(\Sigma,\Sigma')$ for every $\sigma$ and $\sum_\sigma\Sigma'^\sigma=1$, the product group contributes at most $l\,\tilde{K}\,d(\Sigma,\Sigma')+\varepsilon/2$. Altogether the difference is at most $C^*d(\Sigma,\Sigma')+\varepsilon/2\le\varepsilon/2+\varepsilon/2=\varepsilon$, uniformly over $\gamma,\delta,\upsilon$, as claimed. $\blacksquare$