- #1
Mubeena
- 2
- 0
I have a query on a Random process derived from Markov process. I have stuck in this problem for more than 2 weeks.
Let [itex]r(t)[/itex] 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 [itex]i \ne j[/itex], and where [itex]q_{ij}[/itex] is the transition rate and represents the probability per time unit that [itex]r(t)[/itex] makes a transition from state $i$ to a
state $j$. Now, let [itex]r(\rho(t))[/itex] be a random process derived from [itex]r(t)[/itex] depending on a parameter [itex]\rho(t)[/itex], which is defined by
\begin{alignat*}{1}
\frac{d}{dt}\rho(t)=f(r(\rho(t))),\qquad\rho(0)=0
\end{alignat*}
Here [itex]f(.)[/itex] is a piecewise continuous function depending on [itex]r(\rho(t))[/itex]
with range space as [itex]\mathbb{R}[/itex], a set of Real numbers. In this case can we describe the random process [itex]r(\rho(t))[/itex] 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*}
Let [itex]r(t)[/itex] 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 [itex]i \ne j[/itex], and where [itex]q_{ij}[/itex] is the transition rate and represents the probability per time unit that [itex]r(t)[/itex] makes a transition from state $i$ to a
state $j$. Now, let [itex]r(\rho(t))[/itex] be a random process derived from [itex]r(t)[/itex] depending on a parameter [itex]\rho(t)[/itex], which is defined by
\begin{alignat*}{1}
\frac{d}{dt}\rho(t)=f(r(\rho(t))),\qquad\rho(0)=0
\end{alignat*}
Here [itex]f(.)[/itex] is a piecewise continuous function depending on [itex]r(\rho(t))[/itex]
with range space as [itex]\mathbb{R}[/itex], a set of Real numbers. In this case can we describe the random process [itex]r(\rho(t))[/itex] 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*}