Forward Invariance of the Probability Simplex under the Projected Drift

Let $(\beta_0,\beta_1)$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$, let $\beta$ be its projected extension with rate bound $B$, aggregate state drift $b$ and projected drift $\hat{b}$, let $\Delta^l$ be the probability simplex, and let $T>0$ be a real number.

Let $A:[0,T]\to\mathbb{R}^m$ be a map whose components are measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line, and let $x:[0,T]\to\mathbb{R}^l$ be a continuous map with $x_0\in\Delta^l$ such that

The integrands here are measurable and bounded, so the Lebesgue integrals exist; this is verified in the proof.

Then $x_t\in\Delta^l$ for every $t\in[0,T]$, and consequently $\hat{b}(x_t,A_t)=b(x_t,A_t)$ for every $t\in[0,T]$.