I have a query on a Random process derived from Markov process. I have stuck in this problem for more than 2 weeks.
Let $r(t)$ be a finite-state Markov jump process described by \begin{alignat*}{1} \lim_{dt\rightarrow 0}\frac{Pr\{r(t+dt)=j/r(t)=i\}}{dt} & =q_{ij} \end{alignat*} when $i \ne j$, and where $q_{ij}$ is the transition rate and represents the probability per time unit that $r(t)$ makes a transition from state $i$ to a state $j$. Now, let $r(\rho(t))$ be a random process derived from $r(t)$ depending on a parameter $\rho(t)$, which is defined by \begin{alignat*}{1} \frac{d}{dt}\rho(t)=f(r(\rho(t))),\qquad\rho(0)=0 \end{alignat*} Here $f(.)$ is a piecewise continuous function depending on $r(\rho(t))$ with range space as $\mathbb{R}$, a set of Real numbers. In this case can we describe the random process $r(\rho(t))$ as \begin{alignat*}{1} \lim_{dt\rightarrow 0}\frac{\mathrm{Pr}\{r(\rho(t+dt))=j/r(\rho(t))=i\}}{\rho(t+dt)-\rho(t)} =q_{ij},\qquad i\ne j\\ \end{alignat*}
Yes you can but what is asked from you is to note that $\rho(t+s)-\rho(t)=\rho'(t)s+o(s)$ when $s\to0$. Using the ODE defining $\rho$, this is $f(r(\rho(t)))s+o(s)$. Conditioning on $r(\rho(t))=i$, this is $f(i)s+o(s)$. Thus, the process $(x(t))_{t\geqslant0}$ defined by $x(t)=r(\rho(t))$ is a finite-state Markov jump process with transition rates $$ \lim_{s\to0+}\frac{\Pr(x(t+s)=j\mid x(t)=i)}{s}=q_{ij}f(i). $$