Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control

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. Write $K_b=2\sqrt{l}\,(l-1)B$.

Let $S_0\in\Delta^l$ and let $A:[0,T]\to\mathcal{A}$ 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.

1. (Existence.) There is exactly one continuous map $x:[0,T]\to\mathbb{R}^l$ such that

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

Writing $S=x$, one has $S_t\in\Delta^l$ for every $t$, the pair $(S,A)$ is a generalized mean-field trajectory pair for $(\beta_0,\beta_1)$ with horizon $T$ (the drift in condition 2 of that definition is the projected drift, which agrees with $b$ on $\Delta^l$), and the value of $S$ at $t=0$ is $S_0$.

2. (Uniqueness.) If $(\tilde{S},A)$ is a generalized mean-field trajectory pair for $(\beta_0,\beta_1)$ with horizon $T$ whose value at $t=0$ is $S_0$, then $\tilde{S}=S$.

3. (Regularity.) $|S_t-S_r|\le K_b|t-r|$ for all $r,t\in[0,T]$; that is, $S$ is Lipschitz with constant $K_b$.