**Setting**

Let $[0,T]$ with $T\in\mathbb{R}^{+}$ be a time horizon over which $N\in\mathbb{N}^{+}$ continuous-time time-homogeneous Markov chains make transitions between $\{1,...,h\}$ states with $h\in\mathbb{N}^{+}$ and state $h$ being an absorption state.

**Introduction to subject**

(Unnecessary if you are familiar with absorption generator matrices)

First, the Markov and time homogeneity properties naturally lead to the construction of a generator matrix $Q$ with the transition probability matrix over time $P(t)$ being describes as the matrix exponential of $Q\odot t$, such that $P(t)=\text{expm}(Qt)$. The (negative) diagonal elements of $Q$, $q_{i}$, describe the exponentially distributed holding times per state and (nonegative) off-diagonal elements, $q_{ij}$, describe the conditional transition probabilities given the end of a holding time from state $i$ to $j$ with $i,j\in\{1,...,h\}$.

Second, estimating $Q$ through maxmimum likelihood estimation is trivial as $\hat{q}_{ij}=\frac{N_{ij}(T)}{R_{i}(T)}$ with $N_{ij}(T)$ the number of $i\rightarrow j$ transitions and $R_{i}(T)$ is the sum of the holding times in state $i$ on time horizon $T$ for all $N$ Markov chains. Additionally, the likelihood is concave in the parameter space as for every $q_{ij}$, the second derivative of the loglikelihood is non-positive, $\frac{\partial l}{\partial q_{ij}}=-\sum\frac{1}{q_{ij}^{2}}$, summing over the number of $i\rightarrow j$ transitions and $q_{i}=-\sum_{j=1}^{h}q_{ij}$.

**Problem**

Finally, I try to research the following subjects for the MLE of $Q$

- Existence (trivial)
- Uniqueness (loglikehood is concave in every parameter dimension)
- Uniqueness of corresponding $P$, such that it cannot be retrieved by another $Q$

From my understanding, these $3$ conditions are sufficient to claim that the 'true' MLE of $Q$ retrieves a 'true' $P$ (or as close as possible via MLE). However, I run in some trouble for the third statement and I was wondering if the following reasoning holds for generator matrices with an absorption state to prove this

- Assume all eigenvalues of $Q$ are unique and nonnegative ($\det(Q)=0$ due to the absorption state), thus after diagonalization $Q=PDP^{-1}$
- Exponential function is a bijection

$$ P_{true}=\text{expm}(Q_{1})=P_{1}\text{exp}(D_{1})P_{1}^{-1}\neq P_{2}\text{exp}(D_{2})P_{2}^{-1}=\text{expm}(Q_{2})=P_{true}\implies Q_{1}=Q_{2} $$

Thanks!

**Edit**

I think I have an easier solution: if the matrix logarithm of $P$ exists, that means $\log(P)=Q_{1}\neq Q_{2}=\log(P)$ implies $Q_{1}=Q_{2}$. the matrix logarithm exists if P is invertible or $\det(P)\neq0$. As $Q$ is diagonizable by assumption (in case of CTMC models probability of not having diagonizable $Q$ is actually $0$) with $Q=PDP^{-1}$, then $P=Pe^{D}P$ and $\det(P)=\det(e^{D})\neq0$ as none of the eigenvalues of a real generator matrix is $-\infty$.