Forward Equation for a Finite-State Jump System Driven by Poisson Clocks with the Fresh-Start Property

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ and $\Lambda\ge0$ be real numbers, let $E$ be a nonempty finite set (the state space), and let $\mathsf{A}$ be a nonempty finite set (the clock labels) with $|\mathsf{A}|$ elements. Write $\mathcal{B}_{[0,T]}$ for the trace Borel $\sigma$-algebra on $[0,T]$, integrals over compact intervals for the Lebesgue integral over the compact interval (an integral over a degenerate interval $[t,t]$ being read as $0$), $\mathbf{1}_S$ for the function equal to $1$ on a set $S$ and $0$ off it, and $\mathbf{1}\{\cdots\}$ for the indicator of the condition inside the braces. The data are:

(D1) (Clocks.) For each $a\in\mathsf{A}$ a stochastic process $Y^a=(Y^a_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$, every path of which is a counting path. (No law is prescribed here; the probabilistic input is hypothesis (H2) below. It holds, for instance, when the $Y^a$ are independent Poisson clocks with a horizon $R>\Lambda T$ and, for each $r$, the levels $\mathcal{T}^a_r$ defined below are $\mathfrak{F}_r$-measurable and satisfy the clock-reading bound of Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound with past $\mathfrak{F}_r$: that lemma applied at $r$ with $\bar{c}=\Lambda r$ gives $R^-=R-\Lambda r>\Lambda(T-r)$.)

(D2) (Rates and transitions.) For each $a\in\mathsf{A}$ a rate function $g^a:[0,T]\times E\to[0,\Lambda]$ such that $u\mapsto g^a(u,x)$ is measurable on $[0,T]$ with respect to $\mathcal{B}_{[0,T]}$ for every $x\in E$, and a transition map $\phi^a:E\to E$.

(D3) (State process.) A filtration $(\mathfrak{F}_t)_{t\in[0,T]}$ of sub-$\sigma$-algebras of $\mathcal{F}$ (time index restricted to $[0,T]$), an event $\Omega_0\in\mathcal{F}$ with $P(\Omega_0)=1$, and a family $X=(X_t)_{t\in[0,T]}$ of maps $X_t:\Omega\to E$ such that $\{X_t=x\}\in\mathfrak{F}_t$ for all $t\in[0,T]$ and $x\in E$, and such that for every $x\in E$ the map $(u,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\mathbf{1}\{X_u(\omega)=x\}$ is measurable with respect to the product $\sigma$-algebra $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$.

(D4) (Consumed clock times and counters.) For each $a\in\mathsf{A}$ and $t\in[0,T]$ a random variable $\mathcal{T}^a_t:\Omega\to[0,\Lambda t]$ (the consumed clock time) such that for every $\omega\in\Omega_0$ and every $t\in[0,T]$

$$\mathcal{T}^a_t(\omega)=\int_{[0,t]}g^a(u,X_u(\omega))\,du ,$$

where the integrand $u\mapsto\mathbf{1}_{\Omega_0}(\omega)g^a(u,X_u(\omega))=\sum_{x\in E}g^a(u,x)\mathbf{1}_{\Omega_0}(\omega)\mathbf{1}\{X_u(\omega)=x\}$ is, by (D2) and (D3), a $[0,\Lambda]$-valued section of a $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable map, so that the integral exists for every $\omega\in\Omega_0$ and is $0$ at $t=0$. The counters are $N^a_t=Y^a_{\mathcal{T}^a_t}$. (For instance $\mathcal{T}^a_t=\int_{[0,t]}\mathbf{1}_{\Omega_0}g^a(u,X_u)\,du$ qualifies: it is $\mathcal{F}$-measurable by the Tonelli theorem applied on $[0,T]\times\Omega$ to the integrand multiplied by $\mathbf{1}_{[0,t]}(u)$, the integral over $[0,t]$ and the integral over $[0,T]$ of the product being the integrals over $\mathbb{R}$ of the same zero extension, claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval.) The hypotheses are:

(H1) (Pathwise structure on $\Omega_0$.) For every $\omega\in\Omega_0$: the path $t\mapsto X_t(\omega)$ is piecewise constant and right-continuous, that is, there are a count $K$, either $0$ or a natural number, and times $0<t_1<\dots<t_K\le T$ such that the path is constant on $[0,t_1)$, on $[t_k,t_{k+1})$ for $k\in\{1,\dots,K-1\}$, and on $[t_K,T]$ (constant on $[0,T]$ when $K=0$); for $t\in(0,T]$ write $X_{t-}(\omega)$ for the common value of $X_s(\omega)$ over $s\in(t-\varepsilon,t)$ for some $\varepsilon>0$, which exists and does not depend on $\varepsilon$ by this piecewise-constant structure. Each path $t\mapsto N^a_t(\omega)$ is nondecreasing (the integrand of $\mathcal{T}^a$ is nonnegative and every path of $Y^a$ is a counting path), so its left limit $N^a_{t-}=\sup_{s<t}N^a_s$ exists for $t\in(0,T]$, and likewise for the grand total $N_t=\sum_{a\in\mathsf{A}}N^a_t$, whose left limit is $N_{t-}=\sum_aN^a_{t-}$ (the supremum of a finite sum of nondecreasing functions being the sum of the suprema). It is required that for every $t\in(0,T]$: if $N_t=N_{t-}$ then $X_t=X_{t-}$, while if $N_t=N_{t-}+1$ then $X_t=\phi^a(X_{t-})$ for the unique label $a$ with $N^a_t\neq N^a_{t-}$ (the increments $N^b_t-N^b_{t-}$ being nonnegative integers with sum $1$). Nothing is required at times $t$ with $N_t\ge N_{t-}+2$.

(H2) (Fresh start.) For every $r\in[0,T]$, writing $\varrho_r=\Lambda(T-r)$ for the residual horizon, the residual clocks $\hat{Y}^{a,r}_u=Y^a_{\mathcal{T}^a_r+u}-Y^a_{\mathcal{T}^a_r}$ ($u\ge0$, $a\in\mathsf{A}$) satisfy: each $\hat{Y}^{a,r}_u$ is a random variable; for every $a$ and all real $0=u_0<u_1<\dots<u_p\le\varrho_r$ the increments $\hat{Y}^{a,r}_{u_1}-\hat{Y}^{a,r}_{u_0},\dots,\hat{Y}^{a,r}_{u_p}-\hat{Y}^{a,r}_{u_{p-1}}$ are independent, the $q$-th having the Poisson distribution with parameter $u_q-u_{q-1}$; and the family of $\sigma$-algebras consisting of $\mathfrak{F}_r$ together with the $\sigma$-algebras generated $\sigma(\hat{Y}^{a,r}_u:0\le u\le\varrho_r)$, one for each $a\in\mathsf{A}$, is independent.

Then, writing $\mathbb{E}$ for the expectation:

(a) (Forward equation in conditioned form.) For every function $F:E\to\mathbb{R}$, all $0\le r\le t\le T$, and every event $D\in\mathfrak{F}_r$,

$$\mathbb{E}\bigl[\bigl(F(X_t)-F(X_r)\bigr)\mathbf{1}_D\bigr]=\mathbb{E}\Bigl[\mathbf{1}_D\int_{[r,t]}\mathbf{1}_{\Omega_0}\sum_{a\in\mathsf{A}}g^a(u,X_u)\bigl(F(\phi^a(X_u))-F(X_u)\bigr)\,du\Bigr],$$

where the inner integral (read as $0$ when $t=r$) is defined for every $\omega$ and is $\mathcal{F}$-measurable, its integrand being a bounded section of a $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$-measurable map as above.

(b) (Solution of the forward equation.) Let $r\in[0,T)$ and $D\in\mathfrak{F}_r$, and put $\mu^D_u(x)=P(D\cap\{X_u=x\})$ for $u\in[r,T]$ and $x\in E$. For $u\in[0,T]$ and $x\neq y$ in $E$ put $q_u(x,y)=\sum_{a\in\mathsf{A}:\ \phi^a(x)=y}g^a(u,x)$ (an empty sum being $0$). Then $u\mapsto q_u(x,y)$ is measurable on $[0,T]$, $\sum_{y\neq x}q_u(x,y)\le|\mathsf{A}|\Lambda$ for all $u$ and $x$, and $(\mu^D_u)_{u\in[r,T]}$ is a solution of the forward equation on $[r,T]$ for the rates $q$ in the sense of that lemma, applied with the rate bound there taken to be $|\mathsf{A}|\Lambda$ and with its pairing $\mu(F)=\sum_{x\in E}\mu(x)F(x)$.