Regularity and Derivative Bounds of the Extended Aggregate Observation Drift

Let $l$ and $\tilde{l}$ be natural numbers with $l\ge2$ and $\tilde{l}\ge1$, let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, let $(\tilde{U},\bar{\tilde{\beta}})$ be a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}$, and let $\bar{\tilde{b}}$ be the extended aggregate observation drift of $(\tilde{U},\bar{\tilde{\beta}})$. Adopt the coordinate and partial-derivative notation $\partial_\gamma$, $\partial_\delta\partial_\gamma$ of the extension definition, write $\Delta^l$ for the probability simplex, and write $d$ for the Euclidean distance. Then for every $\upsilon\in\{1,\dots,\tilde{l}\}$:

(i) (Regularity, derivative formulas, and restriction.) $\bar{\tilde{b}}^\upsilon$ is of class $C^2$ on the open set $\tilde{U}$; in particular each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ exists and is of class $C^1$ on $\tilde{U}$, and for all $\gamma,\delta\in\{1,\dots,l\}$ and $\Sigma\in\tilde{U}$:

$$\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)=\bar{\tilde{\beta}}(\gamma,\upsilon,\Sigma)+\sum_{\sigma=1}^{l}\Sigma^\sigma\,\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma),$$ $$\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)=\partial_\delta\bar{\tilde{\beta}}(\gamma,\upsilon,\Sigma)+\partial_\gamma\bar{\tilde{\beta}}(\delta,\upsilon,\Sigma)+\sum_{\sigma=1}^{l}\Sigma^\sigma\,\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma).$$

Moreover $\bar{\tilde{b}}$ agrees on $\Delta^l$ with the aggregate observation drift of $\tilde{\beta}$.

(ii) (First-order bounds on the simplex.) $|\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le\tilde{B}+\tilde{K}$ for all $\gamma\in\{1,\dots,l\}$ and all $\Sigma\in\Delta^l$, and

$$|\bar{\tilde{b}}^\upsilon(\Sigma)-\bar{\tilde{b}}^\upsilon(\Sigma')|\le\sqrt{l}\;(\tilde{B}+\tilde{K})\;d(\Sigma,\Sigma')\qquad\text{for all }\Sigma,\Sigma'\in\Delta^l.$$

(iii) (Second-order bounds and uniform continuity on the simplex.) $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le3\,\tilde{K}$ for all $\gamma,\delta\in\{1,\dots,l\}$ and $\Sigma\in\Delta^l$; and for every real $\varepsilon>0$ there is a real $\delta^\circ>0$, which may be chosen independently of $\gamma$, $\delta$, and $\upsilon$, such that $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)-\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma')|\le\varepsilon$ for all $\gamma,\delta\in\{1,\dots,l\}$, all $\upsilon\in\{1,\dots,\tilde{l}\}$, and all $\Sigma,\Sigma'\in\Delta^l$ with $d(\Sigma,\Sigma')\le\delta^\circ$.