Conditionally Gaussian stochastic integral with an independent integrator ($\int HdB\mid H\sim N(0,\int H^{2}dt)$)

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.