Is it true in general that if $f$ is a deterministic function, and $W$ is brownian motion, then the quadratic variation of $\int_0^t f(W_s) dW_s$ is $\int_0^t f^2(W_s) ds$?
Is it also true in general that the quadratic variation of $\int_0^t f(s)dW_s$ is $\int_0^t f^2(s)ds$?
Also, is the quadratic variation of $\int_0^t f(W_s) ds =0$?
I have been using these formulas in my work as they seem to be generally true, but I haven't been able to prove them (I struggle to work with quadratic variation from the definition) and I would love to see a proof.
My definition of quadratic variation of a continuous local martingale $M$ is the unique, continuous, increasing and adapted process $\langle M\rangle$ with $\langle M\rangle _0 =0 $ such that $M^2 - \langle M \rangle$ is a continuous local martingale.
Thanks very much!