= Solution
{c}
A finite-horizon form of <Girsanov theorem> is as follows. If $\theta$ is predictable, $\int_0^T\theta_s^2ds<\infty$ almost surely, and its <stochastic exponential> $M$ is a true <martingale> of mean one, then under the <probability> $d\mathbb Q=M_Td\mathbb P$ the process
$$
B_t-\int_0^t\theta_sds,\qquad0\leq t\leq T,
$$
is <Brownian motion>. The same conclusion holds on the completed natural <filtration (probability theory)>, since a strictly positive density gives equivalent probabilities.
Part (b) establishes precisely the required density <martingale> and integrand $\theta_s=\partial_x\log p(1-s,B_s,y)$. Therefore
$$
\boxed{X_t=B_t-\int_0^t\frac{y-B_s}{1-s}\,ds
\text{ is }\mathbb Q\text{-Brownian on }[0,T].}
$$
In particular it is the requested <local martingale>. Equivalently, under $\mathbb Q$, $B$ solves $dB_t=(y-B_t)(1-t)^{-1}dt+dX_t$, the <Brownian bridge> <drift coefficient> toward $y$ at time one. Keeping $T<1$ avoids the singular endpoint in this density change.
Back to article page
