Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 44 / 4 / d / Solution

The supermartingale conclusion givesDiscard the nonnegative terminal-wealth term and let $t\to \infty$. The cumulative deflated consumption increases, so monotone convergence theorem yields the state-price budget constraintThe normalization is relevant: with $Z_0\ne 1$ the right side is $Z_0{X}_0$; with random integrable initial deflated wealth its expectation is the unconditional budget bound.