= Half-volatility measure for a square-root stock claim
{title2=$Z=\mathcal E(\tfrac12\int\sigma\,dW)$}
For bounded <predictable> $\sigma$ on $[0,T]$, the <stochastic exponential> $Z_s=\exp(\tfrac12\int_0^s\sigma_u\,dW_u-\tfrac18\int_0^s\sigma_u^2du)$ is a positive <martingale> by the <Novikov condition>. Under $dQ/dP=Z_T$, the <Bayes formula for conditional expectation> gives
$$
\mathbb E_P[\sqrt{S_T}\mid\mathcal F_t]=\sqrt{S_t}\,\mathbb E_Q\left[\exp\left(-\frac18\int_t^T\sigma_u^2du\right)\middle|\mathcal F_t\right].
$$
Indeed, the square-root stock ratio is $(Z_T/Z_t)\exp(-\tfrac18\int_t^T\sigma_u^2du)$. This exact identity gives the bounds on <square-root stock implied volatility> by bounding integrated variance pathwise. The <Girsanov theorem> gives $W^Q=W-\tfrac12\int\sigma\,du$. The measure is an auxiliary pricing identity for this payoff, not a claim that it is an <equivalent martingale measure> for the original stock.
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