Put $\theta_t=at^{a-1}$. When $a>1/2$,
$$
\int_0^T\theta_t^2dt
=\frac{a^2}{2a-1}T^{2a-1}<\infty.
$$
The integrand is deterministic, so the <Novikov condition> holds. Define an <equivalent probability measure> $Q$ by
$$
\frac{dQ}{dP}
=\exp\left(-\int_0^T\theta_s\,dW_s-
\frac12\int_0^T\theta_s^2ds\right).
$$
The <Cameron-Martin-Girsanov theorem> makes
$$
W_t^Q=W_t+\int_0^t\theta_sds=W_t+t^a=S_t
$$
a $Q$-Brownian motion on $[0,T]$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 3 / c / Solution

Two <probability measures> are equivalent when each is <absolute continuity of measures>[absolutely continuous] with respect to the other, or equivalently when they have the same null events.

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