I encoutered this problem while self-studying Karatzas&Shreve.I have worked out the equivalence relation between $\Lambda_t$ and $M_t$, which is setting $C_t = e^{-\alpha t} $ and $C_t = e^{\alpha t}$ respectively and using Fubini's. But I can't workout the equivalence between $M_t$(or $\Lambda_t$) and $N_t$.
Here are something I've tried:
(1) I tried to use Ito's Lemma since if we assume $M_t \in {\cal M}^{c,loc}$, then $f(X_t)$ is a semi-martingale. But the exponential term becomes a problem.
(2) If we assume $f(X_0) = 0$ (WLOG), then $f(X_t)\frac{d}{dt}\exp\left\{-\int_0^t \frac{{\cal A}f(X_s)}{f(X_s)}\right\} =\exp\left\{-\int_0^t \frac{{\cal A}f(X_s)}{f(X_s)}\right\}{\cal A}f(X_t)$ which looks slightly better not still going to nowhere.
(3) There is no obvious way to apply the hint.
So right now I am just trying random things to see if any of those leads to a spark. But so far none of then did. I am hoping someone could give me some pointers. Thanks in advance.
Call the exponential $Y_t$, so that $$ dY_t = -{\mathcal A f(X_t)\over f(X_t)}Y_t\,dt. $$ Now use the "product rule", noting that the covariation between $f(X_t)$ and $Y_t$ vanishes because $Y_t$ is of finite variation: $$ \eqalign{ d[f(X_t)Y_t)] &= -\mathcal A f(X_t)Y_t\,dt+Y_tdf(X_t),\cr &=-\mathcal A f(X_t)Y_t\,dt+Y_t\cdot(dM_t+\mathcal A f(X_t)\,dt)\cr &=Y_t\,dM_t.\cr } $$