Consider a generator matrix $Q\in\mathbb{R}^{h\times h}$ for a discrete state space $\{1,...,h\}$. I want to determine the probability of a single transition of a stochastic process $X(t)$ with $X(0)=i$ and $X(\Delta t)=j$ with both $i,j\in\{1,..,h\}$.
The transition probability matrix during timestep $\Delta t$ is found by taking the matrix exponential of the generator matrix, thus $P_{\Delta t,i\rightarrow j}=\exp(Q\cdot \Delta t)_{i,j}$. The transition probability matrix takes into account every possible path from $X(0)=i$ to $X(\Delta t)=j$. $P_{\Delta t,i\rightarrow j}$ includes the probability of $X(t)$ undergoing multiple transitions between $0$ and $\Delta t$ with $X(0)=i$ and $X(\Delta t)=j$, for example $X(0)=i$, $X(\Delta t/2)=h$ and $X(\Delta t)=j$ with $i\neq h\neq j$. The probability of $P_{\Delta t,i\rightarrow j}$ is different than the probability of $X(t)$ undergoing one single transition from $i$ to $j$ for $t\in[0,\Delta t)$.
In the derivation of the generator matrix and corresponding transition probability matrix we have several useful definitions. The stochastic process $X(t)$ with $X(0)=i$ remains in state $i$ for a random amount of time, which is exponentially distributed, thus $\sim q_{i}\exp(-q_{i}\cdot\Delta t)$ with $q_{i}=-q_{ii}$. Given a transition from state $i$ at $t\in[0,\Delta t)$ the conditional probability is defined as $\mathbb{P}(X(t)=j|X(t^{-})=i,X(t)\neq i)=\frac{q_{ij}}{q_{i}}$ with $q_{ij}=\lim_{\Delta t\rightarrow0}\frac{\mathbb{P}(X(\Delta t)=j|X(0)=i)}{\Delta t}$.
My attempt was to simulate for every timestep $\Delta t$ the probability of one single transition of $X(t)$ from state $i$ to $j$ by $q_{i}\exp(-q_{i}\cdot t)\frac{q_{ij}}{q_{i}}\exp(-q_{j}(\Delta t-t))=q_{ij}\exp(-q_{i}\cdot t)\exp(-q_{j}(\Delta t-t))$ for $t\in[0,\Delta t)$. Next, probabilities for all possible values of $t\in[0,\Delta t)$ must be summed. By integrating over $t$, I find $$ \begin{split} \mathbb{P}(X(t^{+}=j),X(t^{-}=i\neq j),t\in[0,\Delta t)|X(0)=i)&=\int^{\Delta t}_{0}q_{ij}e^{-q_{i}\cdot t}e^{-q_{j}(\Delta t-t)}\\ &=\frac{q_{ij}e^{-q_{j}\cdot\Delta t}}{q_{j}-q_{i}}(e^{\Delta t(q_{j}-q_{i})}-1) \end{split} $$ with the probability of $X(t)$ remaining in state $i$ from $0$ to $\Delta t$ equal to $$1-\frac{q_{ij}e^{-q_{j}\cdot\Delta t}}{q_{j}-q_{i}}(e^{\Delta t(q_{j}-q_{i})}-1)$$.
My first question: does this approach correctly reflect the right probabilities?
These strong assumptions are made, because in this simulation only one single transition per timestep $\Delta t$ is possible. Transition times of the simulation are subsequently used to determine the likelihood, given a generator matrix; however, in determining the likelihood every transition is considered to occur at the end of every timestep $\Delta t$, which creates an error.
My second question: what approach or theory can be used to approximate the error in the aforementioned method of determining the likelihood?
Thanks in advance.