I have a Question regarding the chapter 5.5.(Girsanov Theorem) of the Book "Brownian Motion,Martingales, and Stochastic Calculus" from le Gall.
There is stated in Prop 5.21, that if D is a continuous local martingale taking strictly positive values, then exists a continuous local martingale L, s.t.
$D_t = \exp\!\left( L_t - \frac12 \langle L, L\rangle_t \right) = \mathcal{E}(L)_t $
and $L$ is given by the formula
$L_t = \log D_0 + \int_0^t D_s^{-1}\, dD_s .$
Now if we assume that D is a (continuous strictly positive) UI martingale. Can we also proof that L is a martingale and not just a local martingale? If so, how would we do that?
So my idea is so far to argue over the bracket process of $L$. We can argue:
$\langle L \rangle_t = \langle (1/D * D) \rangle_t = (1/D^2 * \langle D \rangle)_t$ Since D is strictly positive and comtinuous on a compact set, the essential infimum is positive. Thus we can estimate the right side by some constant $c * \langle D \rangle_t$. Now we can Apply Theorem 4.13 from Le Gall which implies, $E (\langle D \rangle_t) < \infty$ and therefore also $E(\langle L \rangle_t) $ which implies again with Prop 4.13 from Le Gall, that L is indeed a martingale. But I think this proof is not correct since I do not know if D is indeed L2 bounded.