Before revision:  The latter yields a model under the observation filtration  where the drift process \mathbb{F}  becomes a diffusion driven by the innovations process, an  \mathbb{F} -Brownian motion also driving the stock under  \widehat{\mathbb{F}} , and the partial information optimal stopping problem has two spatial dimensions.
After revision:  The latter yields a model under the observation filtration  where the drift process \mathbb{F} in which the stock drift becomes a diffusion driven by the innovations process, an  \mathbb{F} -Brownian motion also driving the stock under  \widehat{\mathbb{F}} , and the partial information optimal stopping problem has two spatial dimensions.