Problem: Doob-Meyer decomposition for a counting process (Exam 2022)

Let \( \{Z_i\}_{i = 0}^\infty \) be a discrete time Markov chain where \( Z_i \in\{0,1\} \) for all \( i = 1, 2, \dots \) and with \(Z_0 = 0\). Let the transition probabilities be given by \( P(Z_{i+1} = 1 \mid Z_i = 0) = \alpha \) and \( P(Z_{i+1} = 0 \mid Z_i = 1) = \beta \) for all \( i = 0, 1, 2, \dots \).

Let \( \{N(t); t \geq 0\} \) be a counting process which is independent of \( \{Z_i\}_{i = 0}^\infty \) and let \( \lambda(t) \) denote the intensity process of \( N(t) \) . For \( k = 1, 2, \dots \) we let \(T_k\) denote the \( k \)'th event time in \( N(t) \), and set as usual \( T_0 = 0 \) . Define a process \( \{ X(t), t \geq 0\} \) by

\[ X(t) = \sum_{i = 1}^{N(t)} Z_i \text{ for } t \geq 0. \]

Let \( \{\mathcal{F}_t \}\) be the history that at time \( t \) contains all information about the counting process \( N(t) \) up
to and including time \( t \) and information about \( Z_i\), \( i = 1, \dots, N(t) \).

Problem: Explain why it is obvious that \(X(t)\) is a sub-martingale.

Find an expression for the incremental process \( dX(t) \) and use this to derive an expression for
the incremental compensator process

\[dX^*(t) = \text{E}[dX(t) \mid \mathcal{F}_{t-}].\]

Use the expression you found for \( dX^*(t) \) to derive an expression for the compensator \( X^*(t) \).