CTMC Chapman-Kolmogorov Equation

Theorem

For all $s,t\ge 0$ and all states $i,j$:

$P_{ij}(t+s)={\sum }_{k=0}^{\infty }{P}_{ik}(t)P_{kj}(s)$

Interpretation

To go from state $i$ to state $j$ in time $t+s$, the process first goes from $i$ to some intermediate state $k$ in time $t$ (with probability $P_{ik}(t)$), and then from $k$ to $j$ in the remaining time $s$ (with probability $P_{kj}(s)$). We sum over all possible intermediate states $k$.

This is the continuous-time analog of the Chapman-Kolmogorov Equation for discrete-time Markov chains.

Matrix Form

In matrix notation, with $P(t)=(P_{ij}(t))$:

$P(t+s)=P(t)P(s)$

Key Role

The Chapman-Kolmogorov equation is essential for deriving the Kolmogorov Differential Equations (both forward and backward), which govern the evolution of the TPF.

Related

• Chapman-Kolmogorov Equation
• CTMC Transition Probability Function
• Kolmogorov Differential Equations

Exercises

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CTMC 2-state. Misalkan CTMC dengan $S=\{0,1\}$ dan $Q=\left[\begin{array}{cc}-\lambda & \lambda \\ \mu & -\mu \end{array}\right]$. Diketahui $P_{00}(t)=\frac{\mu }{\lambda +\mu }+\frac{\lambda }{\lambda +\mu }{e}^{-(\lambda +\mu )t}$.

Verifikasi persamaan Chapman-Kolmogorov $P_{00}(t+s)=P_{00}(t)P_{00}(s)+P_{01}(t)P_{10}(s)$.

Petunjuk: Gunakan $P_{01}(t)=1-P_{00}(t)$ dan $P_{10}(t)=\frac{\mu }{\lambda +\mu }(1-e^{-(\lambda +\mu )t})$. Substitusi dan sederhanakan.