A Jump Processes \((X_{t})\) is a homogeneous Markov chain with transition probabilities \[ p_{i,j}(s,t):=\mathbb{P}(X_{t}=j|X_{s}=i)=p_{i,j}(t-s);\quad(s<t), \] if for all \(n>1\), for all \(t_{1}<t_{2}<\dots<t_{n}\) and for all \(j,i_{1}, \dots, i_{n-1}\in S\) holds \[ \mathbb{P}(X(t_{n})j|X(t_{1})=i_{1}, \dots , X(t_{n-1})=i_{n-1})=\mathbb{P}(X(t_{n})=j|X(t_{n-1})=i_{n-1})=p_{i_{n-1},j}(t_{n}-t_{n-1}). \] In previous notes we constructed a Markov chain via holding rates \(\{ q_{i},~i \in S \}\) and transition matrix \(P=(p_{i,j})_{i,j \in S}\) for an imbedded jump chain \((X_{n})\).
Assumptions: We now make the following assumptions:
- The Markov chain \((X_{t}, t\geq 0)\) is regular (\(S_{\infty}=\infty\)); and
- All states are stable (\(0<q_{i}<\infty\)).
0.1 Integral Equation for Transition Probabilities
The following integral equation holds for transition probabilities \(p_{i,j}(t)\) \[ p_{i,j}(t)=\delta_{i,j}e^{-q_{i}t}+\int_{0}^tq_{i}e^{-q_{i}s}\sum_{k \neq i}p_{i,k}\cdot p_{k,j}(t-s)ds, \] for all \(t>0\) and \(i,j \in S\).
To provide an interpretation of this result, the LHS of the equation is \(\mathbb{P}(X_{t}=j|X_{0}=i)\) and the RHS depends on \((P,(q_{i}))\), parameters in construction of the chain. ### The Generator and Backward Equations
Firstly we note that \(P_{t}\searrow I\) when \(t \to 0\) since the integral \(\int_{0}^t\) converges to \(0\) when \(t \to 0\), hence \(P_{t}\) is standard.