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

Explaining that \( X(t) \) is a sub-martingale: Use the definition of sub-martingale (2.33 in ABG), and rewrite \( X(t) = X(t) + X(s) - X(s)\) to show the inequality.

Finding the incremental compensator process \(dX^*(t)\) : Consider what values \(Z_i\) and \(dN(t)\) can take.

Finding the compensator process \(X^*(t)\): Remember that \(Z_{N(u−)}\) will be constant between event times \(T_{i-1}\) and \(T_i\), which means we can split the integral into a sum of many integrals.

Solution: To get a detailed solution of the problem, see the link at the bottom of this page.