Terminal nonnegativity criterion for a finite-horizon martingale transform ($Y_0=0,Y_T\ge 0⟹Y_T=0\text{ a.s.}$)

For $Y_t={\sum }_{s\le t}{K}_s(M_s-M_{s-1})$ 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.