Conditional square-root price under independent volatility ($k=1/8$)

In the joint filtration of Brownian history and an independent volatility history, conditioning on the entire volatility path and then using the tower property gives $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\mathbb{E}[\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)\mid {\mathcal{F}}_t]$. Removing the outer conditional expectation requires the integrated variance to be known at time $t$. Independence alone does not give this measurability; a volatility parameter disclosed later supplies a counterexample.