The Projected Extension of an Affine-Controlled Transition-Rate Family

Let $(\beta_0,\beta_1)$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$ and Lipschitz constant $\Lambda$, and let $\Delta^l$ be the probability simplex. Write $\pi_{\mathcal{A}}$ and $\pi_{\Delta^l}$ for the nearest-point projections onto $\mathcal{A}$ and onto $\Delta^l$, and define the projected extension of $(\beta_0,\beta_1)$ by

for each ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ in $\{1,\dots,l\}$. Let

the latter suprema being over all ordered pairs $(\sigma,\gamma)$ with $\sigma\neq\gamma$, all $\Sigma\in\Delta^l$ and all $\alpha\in\mathbb{R}^m$, and set $\Lambda_\beta=\Lambda(1+R)$ and $\Lambda_b=2\sqrt{l}\,(l-1)(B+\Lambda_\beta)$.

1. (Finiteness.) The projections $\pi_{\mathcal{A}}$ and $\pi_{\Delta^l}$ are defined, and $R$, $K_1$, $B$ are finite nonnegative real numbers.

2. (Transition-rate family.) $\beta$ is a transition-rate family on $l$ states with control dimension $m$ and rate bound $B$. Moreover $\beta(\sigma,\gamma,\Sigma,\alpha)=\beta_0(\sigma,\gamma,\Sigma)+\beta_1(\sigma,\gamma,\Sigma)\cdot\alpha$ whenever $\alpha\in\mathcal{A}$, and for each fixed $\alpha\in\mathbb{R}^m$ the map $\Sigma\mapsto\beta(\sigma,\gamma,\Sigma,\alpha)$ is Lipschitz with constant $\Lambda_\beta$ on $\Delta^l$.

3. (Affine drift.) Let $b$ be the aggregate state drift of $\beta$, and define, for $\Sigma\in\Delta^l$ and $\gamma\in\{1,\dots,l\}$,

with values in $\mathbb{R}$ and $\mathbb{R}^m$ respectively. Then

4. (Bounds.) For all $\Sigma,\Sigma'\in\Delta^l$ and $\alpha,\alpha'\in\mathbb{R}^m$,

5. (Conservation and inflow bound.) For all $\Sigma\in\Delta^l$, $\alpha\in\mathbb{R}^m$ and $\gamma$,

6. (Projected drift.) Define $\hat{b}:\mathbb{R}^l\times\mathbb{R}^m\to\mathbb{R}^l$ by $\hat{b}(x,\alpha)=b(\pi_{\Delta^l}(x),\alpha)$. Then $\hat{b}(x,\alpha)=b(x,\alpha)$ for $x\in\Delta^l$, $|\hat{b}(x,\alpha)|\le2\sqrt{l}\,(l-1)B$ for all $x\in\mathbb{R}^l$ and $\alpha\in\mathbb{R}^m$, and $|\hat{b}(x,\alpha)-\hat{b}(x',\alpha)|\le\Lambda_b|x-x'|$ for all $x,x'\in\mathbb{R}^l$ and $\alpha\in\mathbb{R}^m$.