I'm trying to show that for a standard Brownian motion and some twice continuously differentiable function $f$ that $f(B_t)$ is a local martingale iff $f'' = 0$.
Applying Ito's formula gives, and since $\langle B_s\rangle = s$: $$f(B_t) = f(B_0) + \int_0^t f'(B_s) dB_s + \frac{1}{2} \int_0^t f''(B_s) ds$$
Ito's formula tells us that $f(B_t)$ is a semimartingale, so has unique decomposition $$f(B_0) + M_t + A_t$$ where $M_t$ is a local martingale and $A_t$ is a process of finite variation. Presumably we use uniqueness of the decomposition to match up the terms of these expressions and the result follows, so my only questions are why
- $\int_0^t f'(B_s) dB_s$ is a local martingale?
- $\int_0^t f''(B_s) ds$ is a process of finite variation?
Sorry for the long preamble, thank you for any help!