Stopping-time shift of a stochastic integral (${\int }_T^{T+t}{H}_{s}d{M}_s={\int }_0^t{H}_{T+s}d(M_{T+s}-M_T)$)

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.