Transition rate matrix from transition probability matrix If I have a $2 \times 2$ continuous time Markov chain transition probability matrix (generated from a financial time series data), is it possible to get the transition rate matrix from this and if Kolmogorov equations can assist, how would I apply them.
 A: If $P_t=\begin{pmatrix}1-a(t)&a(t)\\ b(t)&1-b(t)\end{pmatrix}$  and $Q=\begin{pmatrix}-\alpha&\alpha\\ \beta&-\beta\end{pmatrix}$ are such that $P_t=\mathrm e^{Qt}$, then $$P'_t=P_tQ,$$ hence 
$$a'(t)=\alpha(1-a(t))-\beta a(t),\qquad b'(t)=\beta(1-b(t))-\alpha b(t).$$ 
Thus, 
$$a'(t)+(\alpha+\beta)a(t)=\alpha,\qquad b'(t)+(\alpha+\beta)b(t)=\beta.$$ Since $a(0)=b(0)=0$, one gets 
$$a(t)=\frac{\alpha}{\alpha+\beta}(1-\mathrm e^{-(\alpha+\beta)t}),\qquad b(t)=\frac{\beta}{\alpha+\beta}(1-\mathrm e^{-(\alpha+\beta)t}).
$$
Finally,
$$
P_t=\frac{1}{\alpha+\beta}\begin{pmatrix}\beta+\alpha\mathrm e^{-(\alpha+\beta)t}&\alpha(1-\mathrm e^{-(\alpha+\beta)t})\\ \beta(1-\mathrm e^{-(\alpha+\beta)t})&\alpha+\beta\mathrm e^{-(\alpha+\beta)t}\end{pmatrix}.
$$
Note that by homogeneity there are only two parameters here, namely $A=\alpha t$ and $B=\beta t$, since
$$
P_t=\frac{1}{A+B}\begin{pmatrix}B+A\mathrm e^{-A-B}&A(1-\mathrm e^{-A-B})\\ B(1-\mathrm e^{-A-B})&A+B\mathrm e^{-A-B}\end{pmatrix}.
$$
If the observed matrix is
$$
P_{\mathrm{obs}}=\begin{pmatrix}1-U& U\\ V&1-V\end{pmatrix},
$$
it is necessary that $U+V\lt1$ (otherwise, $P_{\mathrm{obs}}$ does not correspond to a continuous time transition matrix), then one can estimate $A$ and $B$ by
$$
\hat A=-\log(1-U-V)\cdot\frac{U}{U+V},\qquad \hat B=-\log(1-U-V)\cdot\frac{V}{U+V}.
$$
