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 transition-rate family 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 $A:[0,T]\to\mathcal{A}$ be a map whose components $A^1,\dots,A^m:[0,T]\to\mathbb{R}$ 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 — the interval $[0,T]$ regarded as a subset of the real line with the absolute value metric and $\mathbb{R}^l$ carrying the Euclidean distance — with $x_0\in\Delta^l$ such that

$$x^\gamma_t=x^\gamma_0+\int_{[0,t]}\hat{b}^\gamma(x_s,A_s)\,ds\qquad\text{for all }t\in[0,T]\text{ and all }\gamma\in\{1,\dots,l\} .$$

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 $t\in[0,T]$.