What do we eaxctly mean by the term transition intensity and how is it different from transition probability? Transition Intensity = lim dt-0 d/dt (dtQx+t/dt)
where dtQx+t= P(person in the dead state at age x+t+dt/given in the alive state at age x+t) Dead and alive are just examples it can be from any one state to another
For simplicity, consider a two-state Markov process $(X_t)$ on the states {$\texttt{alive}$,$\texttt{dead}$} with only one transition, from $\texttt{alive}$ to $\texttt{dead}$, with rate transition $r$. This means that, for every $t$, when $s\to0$, $$P(X_{t+s}=\texttt{dead}\mid X_t=\texttt{alive})=rs+o(s).$$ This implies that, for every $s$ and $t$, $$P(X_{t+s}=\texttt{alive}\mid X_t=\texttt{alive})=\mathrm e^{-rs},$$ and $$P(X_{t+s}=\texttt{dead}\mid X_t=\texttt{alive})=1-\mathrm e^{-rs}.$$ To sum up, transition intensities, such as $r$, are derivatives with respect to time at time zero of transition probabilities, such as $1-\mathrm e^{-rs}$. The former can be any nonnegative real number while the latter is always a real number in $[0,1]$.