= Stopped predictable budget for adapted increments
{title2=$A_n=a-\sum_{j<n\wedge\tau_a}Y_j$}
For finite nonnegative <adapted processes> $Y_j$, set $\tau_a=\inf\{n:\sum_{j\leq n}Y_j>a\}$ and $U_n=\sum_{j<n}Y_j$. Stopping this one-step-lagged sum gives $A_n=a-U_{n\wedge\tau_a}\in[0,a]$, with $A_{n+1}-A_n=-Y_n\mathbf1_{\{\tau_a>n\}}$. The latter is nonpositive and <measurable> at time $n$, so $A$ is a nonnegative <supermartingale>. The overshooting increment is excluded, which is essential to nonnegativity.
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