= Solution
First propagate terminal nonnegativity backwards; it is not necessary to assume nonnegative wealth at intermediate dates. Suppose $Y_t\geq0$ and define
$$
A_j=\{|Y_{t-1}|\leq j,\ |K_t|\leq j\}\in\mathcal F_{t-1}.
$$
These events increase to the whole space up to a null set. On $A_j$, both the old value and the coefficient are bounded, so $\mathbf1_{A_j}Y_t$ is <integrable> and
$$
\mathbb E[\mathbf1_{A_j}Y_t\mid\mathcal F_{t-1}]
=\mathbf1_{A_j}Y_{t-1}.
$$
The left side is nonnegative; hence $Y_{t-1}\geq0$ on every $A_j$, and therefore almost surely. Starting from $t=T$, induction gives $Y_t\geq0$ for every $0\leq t\leq T$.
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:
$$
\boxed{Y_T=0\quad\text{almost surely}.}
$$
This is the <terminal nonnegativity criterion for a finite-horizon martingale transform>. A finite deterministic horizon is essential to the backward induction.
Back to article page
