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The Girsanov theorem construction proves a weak stochastic solution, but does not by itself establish a strong stochastic solution. It constructs $X=x+W$ first, and then constructs a new driving Brownian motion $B=W-\int b(X_s)ds$. Adaptation of $X$ to the original filtration of $W$ does not by itself prove adaptation to the smaller natural filtration of $B$. Inverting this relation requires an additional result; uniqueness in law alone is insufficient in general.

For the particular coefficients in part (b), however, the literal claim that a strong solution may fail is incorrect. The strong existence theorem for additive-noise SDEs with bounded measurable drift gives strong existence and pathwise uniqueness for bounded Borel state-dependent $b$ and unit diffusion. Consequently every weak realization here is determined by its driving noise and initial value. This theorem is deeper than the Girsanov theorem argument and no proof is needed here. Examples with path-dependent drift or with a nonconstant diffusion coefficient do not contradict it.

The valid distinction is therefore: the argument in part (b) has established weak existence and uniqueness in law; strong existence requires an additional theorem, and in this class that theorem is available. A primary reference for the correction is On strong solutions and explicit formulas for solutions of stochastic integral equations.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.