= Terminal nonnegativity criterion for a finite-horizon martingale transform
{title2=$Y_0=0,\ Y_T\geq0\ \Longrightarrow\ Y_T=0\text{ a.s.}$}
For $Y_t=\sum_{s\leq t}K_s(M_s-M_{s-1})$ and fixed finite deterministic $T$, bounded events in $\mathcal F_{t-1}$ restricting both $Y_{t-1}$ and $K_t$ show that nonnegativity of $Y_t$ forces nonnegativity of $Y_{t-1}$. Induction makes the entire stopped process nonnegative. The <nonnegative discrete-time local martingale is a martingale> criterion then gives $\mathbb E Y_T=Y_0=0$. Nonnegative terminal gain is therefore zero almost surely, even without an intermediate wealth bound.
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