= Predictable-coefficient localization of a martingale transform
{title2=$\sigma_j=\inf\{t\geq0:|K_{t+1}|>j\}$}
For a finite-valued <predictable process> $K$, this is a <stopping time> and increases to infinity. The stopped <martingale transform> uses coefficients $K_s\mathbf1_{\{s\leq\sigma_j\}}$, which are <predictable> and bounded by $j$. Thus a transform of a discrete-time <martingale> is a <local martingale> without a global boundedness assumption. The stop is before, not after, the first large coefficient is used.
Back to article page
