Let $a$ be some real number and $B_t$ a Brownian motion. I want to compute the Doob decomposition of $\cos(aB_t)$, and the predictable variation of the local martingale part $M_t$.
The Doob decomposition is given by:
where $A_t$ is a continuous finite variation process and $M_t$ is a continuous local martingale.
I believe the left term is the $A_t$ and the right $M_t$, but how to compute the predictable variation?