Two books construct Markov processes from Q-matrices using waiting times and jump chains but differ in whether the waiting times depend on the current state. Can the two be reconciled?
Klenke claims that to every Q-matrix $q$ corresponds a unique Markov process that is differentiable at time $t=0$ and such that $q$ is its derivative there (Theorem 17.25). According to him, this unique process can be constructed by combining a (discrete time) Markov chain representing "jumps" between states, and a Poisson process representing the time lapse between jumps (a.k.a. the waiting time). According to this model, the waiting time $\sim \exp(\lambda)$, where $\lambda$ is a function of $Q$, and therefore unaffected by the current state.
Norris too constructs a Markov process from a Q-matrix (section 2.6). His construction is similar to Klenke's except that $\lambda$ is a function not only of $Q$, but of the state active at the beginning of the waiting time.
Can the two constructions be reconciled? What am i missing here? Is one of the authors wrong or idiosyncratic in his definition of a Markov process?
Klenke
Theorem 17.25 Let $q$ be an $E\times E$ matrix ($E$ being a countable set) such that $q(x,y)\geq0$ for all $x,y\in E$ with $x\neq y$. Assume that the following hold
i) $q(x,y)\geq0$ for all $x,y\in E$,
ii) $q(x,x)=-\sum_{y\neq x}q(x,y)$
iii) $\sup_{x\in E}|q(x,x)|<\infty$
Then $Q$ is the $Q$-matrix of a unique Markov process $X$.
Proof [Abridged] Let $I$ be the unit matrix on $E$. Define $p(x,y)=\frac{1}{\lambda}q(x,y)+I(x,y)\space\space\mathrm{for\, }x,y\in E$
Then $p$ is a stochastic matrix and $q=\lambda(p-I)$. Let $\left((Y_n)_{n\in\mathbb{N}_0}, (\mathrm{P}_x^Y)_{x\in E}\right)$ be a discrete Markov chain with transition matrix $p$ and let $\left((T_t)_{t\geq0}, (\mathrm{P}_n^T)_{n\in \mathbb{N}_0}\right)$ be a Poisson process with rate $\lambda$. Let $X_t:=Y_{T_t}$ and $\mathrm{P}_x=\mathrm{P}_x^Y\otimes\mathrm{P}_0^T$. The $\mathfrak{X}:=\left((X_t)_{t\geq0}, (\mathrm{P}_x)_{x\in E}\right)$ is [...] the required Markov process. [...] $\square$
Norris
A minimal right-continuous process $(X_t)_{t\geq0}$ on $I$ is a Markov chain with initial distribution $\lambda$ and generator matrix $Q$ if its jump chain $(Y_n)_{n\geq0}$ is discrete-time Markov($\lambda$, $\Pi$) and if for each $n\geq1$, conditional on $Y_0, \dots, Y_{n-1}$, its holding times $S_1, \dots, S_n$ are independent exponential random variables of parameters $q(Y_0), \dots, q(Y_{n-1})$ respectively. We say $(X_t)_{t\geq0}$ is Markov($\lambda$, $Q$) for short. We can construct such a process as follows: let $(Y_n)_{n\geq0}$ be discrete-time Markov($\lambda$, $\Pi$) and let $T_1,T_2,\dots $ be independent exponential random variables of parameter $1$, independent of $(Y_n)_{n\geq0}$. Set $S_n=T_n/q(Y_{n-1})$, $J_n=S_1+\cdots+S_n$ and $X_t=\begin{cases}Y_n &\mathrm{if\, }J_n\leq t
Then $(X_t)_{t\geq0}$ has the required properties.