= Stopping-time shift of a stochastic integral
{title2=$\int_T^{T+t}H_s\,dM_s=\int_0^tH_{T+s}\,d(M_{T+s}-M_T)$}
For a finite <stopping time> $T$, use the shifted filtration $\mathcal G_s=\mathcal F_{T+s}$. An <martingale> with a bounded $L^2$ norm shifts to the <martingale> $M_{T+s}-M_T$; a continuous bounded adapted integrand shifts to a predictable integrand. Their <quadratic variation> and integration intervals shift by subtracting the value at $T$. Both integrals have identical left-endpoint sums, and the <Itô isometry> passes the equality to the limit. The starting jump at $T$ is excluded.
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