Work first under <Wiener measure> $P$ with coordinate <Brownian motion> $X$. Boundedness of $b$ implies the <Novikov condition>, so
$$
Z_T=\exp\!\left(\int_0^Tb(X_s)dX_s-\frac12\int_0^Tb(X_s)^2ds\right)
$$
has expectation one. Define $Q$ by $dQ=Z_TdP$. The <Girsanov theorem> makes
$$
W_t=X_t-\int_0^tb(X_s)ds
$$
a $Q$-Brownian motion, and hence $(X,W,Q)$ is a <weak solution of a stochastic differential equation>.

For <uniqueness in law>, start with any weak solution under $Q$ and apply the inverse <change of measure> with density $\mathcal E(-\int b(X_s)dW_s)_T$. Boundedness again gives the <Novikov condition>, and under the resulting measure $P$ the process $X$ is Brownian. Reversing the density expresses the law of $X$ under $Q$ as the same functional $Z_T$ of a Wiener path. It is therefore independent of the chosen weak solution. This proves the <Weak existence and uniqueness in law for an additive-noise SDE with bounded drift>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 5 / b / Solution

{c}

If $b:\mathbb R\to\mathbb R$ is bounded and measurable, then on every finite time interval
$$
dX_t=b(X_t)dt+dW_t
$$
has a weak solution and <uniqueness in law>. Starting with <Wiener measure>, the <Novikov condition> and <Girsanov theorem> add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.

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 Incoming links: Weak existence and uniqueness in law for an additive-noise SDE with bounded drift