= Conditionally Gaussian stochastic integral with an independent integrator
{title2=$\int H\,dB\mid H\sim N(0,\int H^2\,dt)$}
If the entire integrand path is independent of the integrating <Brownian motion>, conditioning on that path gives a deterministic square-integrable integrand. The <Itô isometry> and Gaussian approximation by simple integrands show that its <stochastic integral> is conditionally centered normal with the stated variance. A strictly positive finite conditional variance implies that the unconditional distribution has no atoms.
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