= Solution
Under the corrected hypothesis, $N_t=X_t-1$ is a <continuous local martingale> with $N_0=0$, $[N]_t=A_t=\int_0^te^{2B_s}ds$, and $A_\infty=\infty$ <almost surely>. The <Dambis-Dubins-Schwarz theorem> states that, for such a process,
$$
\tau_u=\inf\{t\geq0:[N]_t>u\},\qquad W_u=N_{\tau_u}
$$
defines a standard <Brownian motion> relative to the <filtration> $\mathcal F_{\tau_u}$, and $N_t=W_{[N]_t}$. The divergence of the <quadratic variation> ensures every $\tau_u$ is finite, so no extension of the <probability space> is needed. Here $A$ is continuous and strictly increasing because its derivative is $e^{2B_t}>0$, so the inverse is particularly straightforward. Consequently,
$$
\boxed{X_t=1+W_{\int_0^te^{2B_s}ds}.}
$$
The constructed <Brownian motion> need not be independent of its random clock. Under the literal PDF assumptions $X$ need not be a <local martingale>, as the counterexample in part (a) shows, so this <Dambis-Dubins-Schwarz theorem> conclusion cannot be asserted without the correction.
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