Twice Continuously Differentiable Extension of a Transition-Rate Family

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 $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, and let $K\ge0$ be a real number. Points of $\mathbb{R}^l\times\mathbb{R}^m$ are written $x=(\Sigma,\alpha)$ and identified with points of $\mathbb{R}^{l+m}$, with coordinates $x_1,\dots,x_{l+m}$, so that $x_\gamma=\Sigma^\gamma$ for $\gamma\le l$ and $x_{l+j}=\alpha^j$ for $j\le m$; a product $U'\times V'$ of sets $U'\subseteq\mathbb{R}^l$ and $V'\subseteq\mathbb{R}^m$ is regarded as a subset of $\mathbb{R}^{l+m}$ under this identification. For a real-valued function $f$ on an open subset of $\mathbb{R}^{l+m}$ and indices $i,j\in\{1,\dots,l+m\}$ we write $\partial_i f$ for the partial derivative of $f$ with respect to the $i$th variable, that is, with respect to the coordinate $x_i$, which is unambiguous wherever it exists by Uniqueness of the Partial Derivative on a Euclidean Open Set, and $\partial_j\partial_i f$ for the iterated partial derivative in the sense of clause 4 of that definition, namely $\partial_j$ applied to the function $\partial_i f$.

A triple $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of the transition-rate family $\beta$ with derivative bound $K$ if it consists of an open, convex, bounded set $U\subseteq\mathbb{R}^l$ with $\Delta^l\subset U$, an open, convex, bounded set $V\subseteq\mathbb{R}^m$ with $\mathcal{A}\subseteq V$, and a family of functions $\bar{\beta}(\sigma,\gamma,\cdot,\cdot):U\times V\to\mathbb{R}$ on the product $U\times V$, indexed by the ordered pairs $(\sigma,\gamma)$ with $\sigma,\gamma\in\{1,\dots,l\}$ and $\sigma\neq\gamma$, such that for every such pair:

1. (Extension.) $\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=\beta(\sigma,\gamma,\Sigma,\alpha)$ for all $(\Sigma,\alpha)\in\Delta^l\times\mathcal{A}$.

2. (Regularity.) $\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ is of class $C^2$ on $U\times V$, which is an open subset of $\mathbb{R}^{l+m}$ by claim 2 of Products of Euclidean Open Sets are Open, $U$ and $V$ being open.

3. (Derivative bounds.) $|\partial_i\bar{\beta}(\sigma,\gamma,x)|\le K$ and $|\partial_j\partial_i\bar{\beta}(\sigma,\gamma,x)|\le K$ for all $i,j\in\{1,\dots,l+m\}$ and all $x\in U\times V$.

4. (Uniform continuity of second derivatives.) For every real $\varepsilon>0$ there is a real $\delta>0$ such that $|\partial_j\partial_i\bar{\beta}(\sigma,\gamma,x)-\partial_j\partial_i\bar{\beta}(\sigma,\gamma,y)|\le\varepsilon$ for all $i,j\in\{1,\dots,l+m\}$ and all $x,y\in U\times V$ whose Euclidean distance satisfies $d(x,y)\le\delta$.