transition rates from transition probabilities

Question

Let $(X_t)_{t\in \mathbb Z}$ be a homogeneous Markov chain on some state space $S$ with transition prbabilities $P=[P_{ij}]_{i,j\in S}$. The law $p_i(t):=\mathbb P(X_{t}=i)$ of $X_t$ satisfies: \begin{equation} p_i(t+1)=\sum_{j\in S}\mathbb P(\;X_{t+1}=i\;|\;X_t=j\;)\;p_j(t)=\sum_{j\in S}P_{ij}\;p_j(t) \end{equation} What is the best approximating rate equation for this? \begin{equation} \frac{d}{dt}\tilde p_i(t)=(W\cdot \tilde p(t))_i\; \end{equation} i.e. what is $W$ in terms of $P$, such that the error between $p$ and $\tilde p$ is minimized? For instance I know that \begin{equation} \tilde p(t+\Delta t)=\text{e}^{W\Delta t}\tilde p(t) \end{equation} Thus I need to solve $P=\text{e}^{W}$ for $\Delta t=1$, or in other words $W=\log P$. Is this exact, in the sense that $\tilde p(t)=p(t)$ for $t\in \mathbb Z$ when $W=\log P$?