Problem: Doop decomposition of a partial sum

Let \(X_0 = 0\) and \(X_n = U_1 + \ldots + U_n\) for \(n = 1, 2,\ldots\), where \(U_1, U_2,\ldots\) are independent and identically distributed stochastic variables with \(E[U_i ] = \mu\).

Find the Doob decomposition of the process \(X\). In particular identify the predictable and the innovation part of the \(X\) process.

Hints: To get some hints for how to solve the problem see the link at the bottom of this page.