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The above question was a typo due to the author -- the expression should be evaluated as \begin{equation} E(t|\mathcal{F}_{s}^{W}) = t \end{equation} due to the reasoning in the question. Sorry for the noise.


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Could you please be more specific with your question and post the text here? This will be more helpful for other people visiting the site. Now as far as to where the 1/2 went, usually people put 1/2 in front of the second order term because this will simplify to 1 after the derivation: $$ \frac{\partial x^2}{\partial x} = 2x $$ vs $$ \frac{1}{2} \cdot ...



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