= Conditional square-root price under independent volatility
{title2=$k=1/8$}
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 $\mathbb E[\sqrt{S_T}\mid\mathcal F_t]=\sqrt{S_t}\mathbb E[\exp(-\tfrac18\int_t^T\sigma_u^2du)\mid\mathcal F_t]$. 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.
Back to article page
