Generalized Mean-Field Trajectory Pair

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 $b$ be the aggregate state drift of its transition-rate family, with rate bound $B$, let $\Delta^l$ be the probability simplex, and let $T>0$ be a real number, called the horizon.

A generalized mean-field trajectory pair for $(\beta_0,\beta_1)$ with horizon $T$ is a pair $(S,A)$ of maps $S:[0,T]\to\Delta^l$ and $A:[0,T]\to\mathcal{A}$, with values written $S_t=(S^1_t,\dots,S^l_t)$ and $A_t=(A^1_t,\dots,A^m_t)$, such that:

1. (Regularity.) Every component $t\mapsto S^\gamma_t$ is continuous on $[0,T]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric, and every component $t\mapsto A^j_t$ is measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line.

2. (Dynamics.) For every $\gamma\in\{1,\dots,l\}$ and every $t\in[0,T]$,

$$S^\gamma_t=S^\gamma_0+\int_{[0,t]}b^\gamma(S_s,A_s)\,ds ,$$

the integral being the Lebesgue integral over the compact interval $[0,t]$, and $0$ for $t=0$.

Under condition 1 the integrand is measurable and bounded by $2\sqrt{l}(l-1)B$, so the integral exists; this follows from measurability of continuous functions, measurability of sequentially continuous functions of measurable Euclidean maps, and the bounds of the lemma on affine-controlled data.