Regularity and Derivative Bounds of the Extended Aggregate State Drift

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, let $(U,V,\bar{\beta})$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K$, and let $\bar{b}$ be the extended aggregate state drift of $(U,V,\bar{\beta})$. Adopt the coordinate and partial-derivative notation $\partial_i$, $\partial_j\partial_i$ of the extension definition, write $\Delta^l$ for the probability simplex, write $d$ for the Euclidean distance, and for $i\in\{1,\dots,l+m\}$ and $\sigma\in\{1,\dots,l\}$ set $\delta_{i\sigma}=1$ if $i=\sigma$ and $\delta_{i\sigma}=0$ otherwise. Then for every $\gamma\in\{1,\dots,l\}$:

(i) (Regularity, derivative formulas, and restriction.) $\bar{b}^\gamma$ is of class $C^2$ on $U\times V$, which is an open subset of $\mathbb{R}^{l+m}$ as noted in clause 2 of the extension definition; in particular each $\partial_i\bar{b}^\gamma$ exists and is of class $C^1$ on $U\times V$, and for all $i,j\in\{1,\dots,l+m\}$ and $x=(\Sigma,\alpha)\in U\times V$:

$$\partial_i\bar{b}^\gamma(x)=\sum_{\sigma:\sigma\neq\gamma}\Big(\delta_{i\sigma}\,\bar{\beta}(\sigma,\gamma,x)+\Sigma^\sigma\,\partial_i\bar{\beta}(\sigma,\gamma,x)-\delta_{i\gamma}\,\bar{\beta}(\gamma,\sigma,x)-\Sigma^\gamma\,\partial_i\bar{\beta}(\gamma,\sigma,x)\Big),$$ $$\partial_j\partial_i\bar{b}^\gamma(x)=\sum_{\sigma:\sigma\neq\gamma}\Big(\delta_{i\sigma}\,\partial_j\bar{\beta}(\sigma,\gamma,x)+\delta_{j\sigma}\,\partial_i\bar{\beta}(\sigma,\gamma,x)+\Sigma^\sigma\,\partial_j\partial_i\bar{\beta}(\sigma,\gamma,x)-\delta_{i\gamma}\,\partial_j\bar{\beta}(\gamma,\sigma,x)-\delta_{j\gamma}\,\partial_i\bar{\beta}(\gamma,\sigma,x)-\Sigma^\gamma\,\partial_j\partial_i\bar{\beta}(\gamma,\sigma,x)\Big).$$

Moreover $\bar{b}$ agrees on $\Delta^l\times\mathcal{A}$ with the aggregate state drift of $\beta$.

(ii) (First-order bounds.) $|\partial_i\bar{b}^\gamma(x)|\le l\,(B+K)$ for all $i\in\{1,\dots,l+m\}$ and all $x\in\Delta^l\times\mathcal{A}$; and if in addition $\mathcal{A}$ is convex, then

$$|\bar{b}^\gamma(x)-\bar{b}^\gamma(y)|\le \sqrt{l+m}\;l\,(B+K)\;d(x,y)\qquad\text{for all }x,y\in\Delta^l\times\mathcal{A}.$$

(iii) (Second-order bounds and uniform continuity.) $|\partial_j\partial_i\bar{b}^\gamma(x)|\le 3\,l\,K$ for all $i,j\in\{1,\dots,l+m\}$ and $x\in\Delta^l\times V$; and for every real $\varepsilon>0$ there is a real $\delta>0$ such that $|\partial_j\partial_i\bar{b}^\gamma(x)-\partial_j\partial_i\bar{b}^\gamma(y)|\le\varepsilon$ for all $i,j\in\{1,\dots,l+m\}$, all $\gamma\in\{1,\dots,l\}$, and all $x,y\in\Delta^l\times V$ with $d(x,y)\le\delta$.