DES - Definition of 'Transition Function Matrix for CTHMC'

Transition function Matrix:

Given a Continuous Time Homogeneous Markov Chain, the transition function is defined as follows:

$$p_{i,j}(t)\triangleq P(X(s+t)=j\mid X(s)=i)$$

• Independent of $s$ (for the homogeneity).

These transition functions are arranged in the matrix:

$$H(t)\triangleq {\left[p_{i,j}(t)\right]}_{i,j\in \mathcal{X}}=\left[\begin{array}{ccccc}{p}_{1,1}(t)& p_{1,2}(t)& p_{1,3}(t)& \dots & \\ p_{2,1}(t)& p_{2,2}(t)& p_{2,3}(t)& \dots & \\ p_{2,1}(t)& p_{2,2}(t)& p_{3,3}(t)& \dots & \\ ⋮& ⋮& ⋮& \ddots & \end{array}\right]$$

Properties of the Matrix $H(t)$:

• $H(0)=I,$ where $I$ is the Identity Matrix. [Dim]
• The sum along any row of $H(t)$ is $1$. [Dim]

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~Example: