First we start with a Brownian Motion $B(t)$, $a\leq t\leq b$ and an admissible filtration for the BM $\{\mathcal F_t\}$.
Let $f(t,\omega)$ be a stochastic process satisfying:
$f(t)$ is adapted to the filtration $\{\mathcal F_t\}$
$\int_a^b |f(t)|^2 dt <\infty $ a.s.
Denote the space of all stochastic processes satisfying the latter $\mathcal L_{ad}(\Omega,L^2[a,b])$.
Then we have that
Let $f\in \mathcal L_{ad}(\Omega,L^2[a,b])$ then $$X_t=\int_a^t f(s)dB(s), a\leq t\leq b$$ is a local martingale w.r.t to the filtration $\{\mathcal F_t\}$.
Nothing is further added about this theorem in my textbook , and I would like to understand the proof of the later, in particular I don't get how to find a localising sequence of stopping times.
There's one example that claims:
Let $f(t)=e^{B(t)^2}\in\mathcal L_{ad}(\Omega,L^2[a,b]) $,
Then $$X_t=\int_a^t e^{B(s)^2} dB(s)$$ is a local martingale.
This is compatible with the theorem above but I am not sure how to properly prove this statement.
Any hint will be greatly appreciated.
This is not exactly an elementary proof. Assuming that you are well versed in the properties of the Ito integral for the class of integrands $\{f: E[\int_a^b f^2(s)\, ds] < \infty \}, $ and know how to prove the existence of a continuous martingale modification for the stochastic integral for this class, then the localizing sequence (do you know the exact definition of a localizing sequence here?) can be taken to be $$\tau_n =\inf\bigl\{s: \int_a^s f^2(\omega, t)\, dt > n\} \hskip 4pt \text{ and } \hskip 4pt \tau_n = b \hskip 4pt \text{ if } \hskip 4pt \int_a^b f^2(\omega, t)\, dt \le n $$
Check Steele's Stochastic Calculus and Financial Applications, instead.Or some other textbook on stochastic calculus that is still at a reasonable introductory level.