Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 1 / e / Solution

First propagate terminal nonnegativity backwards; it is not necessary to assume nonnegative wealth at intermediate dates. Suppose $Y_t\ge 0$ and defineThese events increase to the whole space up to a null set. On $A_j$, both the old value and the coefficient are bounded, so $1_{A_j}{Y}_t$ is integrable andThe left side is nonnegative; hence $Y_{t-1}\ge 0$ on every $A_j$, and therefore almost surely. Starting from $t=T$, induction gives $Y_t\ge 0$ for every $0\le t\le T$.

The process stopped at $T$ is now a nonnegative discrete-time local martingale. Part (b) makes it a martingale with initial value zero. Consequently $\mathbb{E}{Y}_T=0$, and a nonnegative random variable with zero expectation vanishes almost surely:This is the terminal nonnegativity criterion for a finite-horizon martingale transform. A finite deterministic horizon is essential to the backward induction.