Uniqueness for the Forward Equation of a Bounded Jump-Rate Family on a Finite Set

Let $\mathbb{R}$ be the real numbers, let $E$ be a nonempty finite set, let $T>0$ and $\Lambda\ge0$ be real numbers, and let $r\in[0,T)$. For $u\in[0,T]$ and $x,y\in E$ with $x\neq y$ let $q_u(x,y)\ge0$ be real numbers such that, for all such $x,y$, the map $u\mapsto q_u(x,y)$ is measurable on $[0,T]$ with respect to the trace Borel $\sigma$-algebra $\mathcal{B}_{[0,T]}$ (below, $\mathcal{B}_{[r,T]}$ denotes the trace Borel $\sigma$-algebra on $[r,T]$), and such that

$$\sum_{y\in E,\,y\neq x}q_u(x,y)\le\Lambda\qquad\text{for all }u\in[0,T]\text{ and }x\in E.$$

For a function $F:E\to\mathbb{R}$ and $u\in[0,T]$ define $\mathcal{L}_uF:E\to\mathbb{R}$ by

$$\mathcal{L}_uF(x)=\sum_{y\in E,\,y\neq x}q_u(x,y)\bigl(F(y)-F(x)\bigr)\qquad(x\in E).$$

For a function $\mu:E\to[0,\infty)$ and $F:E\to\mathbb{R}$ write $\mu(F)=\sum_{x\in E}\mu(x)F(x)$.

A family $(\mu_u)_{u\in[r,T]}$ of functions $\mu_u:E\to[0,\infty)$ is called a solution of the forward equation on $[r,T]$ (for the rates $q$) if:

(i) for every $x\in E$ the map $u\mapsto\mu_u(x)$ is bounded and measurable on $[r,T]$ with respect to $\mathcal{B}_{[r,T]}$; and

(ii) for every $F:E\to\mathbb{R}$ and every $t\in[r,T]$,

$$\mu_t(F)=\mu_r(F)+\int_{[r,t]}\mu_u(\mathcal{L}_uF)\,du,$$

the Lebesgue integral over the compact interval $[r,t]$ (read as $0$ when $t=r$) of the map $u\mapsto\mu_u(\mathcal{L}_uF)=\sum_{x\in E}\mu_u(x)\,\mathcal{L}_uF(x)$, which is bounded and measurable on $[r,T]$ under (i) and the rate hypotheses by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the restriction to $[r,T]$ of a $\mathcal{B}_{[0,T]}$-measurable map being $\mathcal{B}_{[r,T]}$-measurable (its preimages are intersections of members of $\mathcal{B}_{[0,T]}$ with $[r,T]$), and a bounded measurable function on $[r,t]$ being integrable there since the interval has finite measure.

Then: if $(\mu_u)_{u\in[r,T]}$ and $(\nu_u)_{u\in[r,T]}$ are two solutions of the forward equation on $[r,T]$ for the same rates $q$ with $\mu_r=\nu_r$, then $\mu_t=\nu_t$ for every $t\in[r,T]$.