The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data

Let $(\beta_0,\beta_1)$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda$, and let $\Delta^l$ be the probability simplex. Define

$$\beta(\sigma,\gamma,\Sigma,\alpha)=\beta_0(\sigma,\gamma,\Sigma)+\beta_1(\sigma,\gamma,\Sigma)\cdot\alpha\qquad(\Sigma\in\Delta^l,\ \alpha\in\mathcal{A}),$$

for each ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ in $\{1,\dots,l\}$, where $x\cdot y$ denotes the dot product and $|\cdot|$ the Euclidean norm. Let

$$R=\sup_{\alpha\in\mathcal{A}}|\alpha|,\qquad K_1=\sup|\beta_1(\sigma,\gamma,\Sigma)|,\qquad B=\sup\beta(\sigma,\gamma,\Sigma,\alpha),$$

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

1. (Finiteness.) $R$, $K_1$ and $B$ are finite nonnegative real numbers, and the nearest-point projection $\pi_{\Delta^l}$ onto $\Delta^l$ is defined.

2. (Transition-rate family.) $\beta$ is a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$. Moreover, for each fixed $\alpha\in\mathcal{A}$ 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 $\gamma\in\{1,\dots,l\}$,

$$b^\gamma_0(\Sigma)=\sum_{\sigma\neq\gamma}\big(\Sigma^\sigma\beta_0(\sigma,\gamma,\Sigma)-\Sigma^\gamma\beta_0(\gamma,\sigma,\Sigma)\big),\qquad b^\gamma_1(\Sigma)=\sum_{\sigma\neq\gamma}\big(\Sigma^\sigma\beta_1(\sigma,\gamma,\Sigma)-\Sigma^\gamma\beta_1(\gamma,\sigma,\Sigma)\big),$$

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

$$b^\gamma(\Sigma,\alpha)=b^\gamma_0(\Sigma)+b^\gamma_1(\Sigma)\cdot\alpha\qquad\text{for all }\Sigma\in\Delta^l,\ \alpha\in\mathcal{A}.$$

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

$$|b(\Sigma,\alpha)|\le2\sqrt{l}\,(l-1)B,\qquad |b(\Sigma,\alpha)-b(\Sigma',\alpha)|\le\Lambda_b\,|\Sigma-\Sigma'|,\qquad |b(\Sigma,\alpha)-b(\Sigma,\alpha')|\le2\sqrt{l}\,(l-1)K_1|\alpha-\alpha'| .$$

5. (Conservation and inflow bound.) For all $\Sigma\in\Delta^l$, $\alpha\in\mathcal{A}$ and $\gamma\in\{1,\dots,l\}$,

$$\sum_{\gamma=1}^lb^\gamma(\Sigma,\alpha)=0,\qquad b^\gamma(\Sigma,\alpha)\ge-(l-1)B\,\Sigma^\gamma .$$

6. (Projected drift.) Define $\hat{b}:\mathbb{R}^l\times\mathcal{A}\to\mathbb{R}^l$ 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 $\alpha\in\mathcal{A}$, and $|\hat{b}(x,\alpha)-\hat{b}(x',\alpha)|\le\Lambda_b|x-x'|$ for all $x,x'\in\mathbb{R}^l$ and $\alpha\in\mathcal{A}$.