The original question is
Solve the stochastic differential equation $dY_t=rd_t+αY_tdB_t$ where $B_t$ is a Brownian motion.
In Theoretical's answer,
\begin{align*} \mathrm d(FY)_t &= F_t\,\mathrm d Y_t + Y_t\, \mathrm dF_t + \mathrm d [F,Y]_t \\ &= r F_t\,\mathrm d t + \alpha F_t Y_t \,\mathrm d B_t +\alpha^2 F_t Y_t \,\mathrm dt -\alpha F_t Y_t\,\mathrm d B_t -\alpha^2 F_t Y_t \,\mathrm d t \\ &=r F_t \,\mathrm dt. \end{align*}
how can we calculate $d[F,Y]_t$ in the second equation? I've done some search and find that it should be $o(\sqrt{(dF_t)^2+(dY_t)^2})$ in ODE but I have no idea about it in SDE.
Thanks a lot for answering.
Thanks to Kurt, I searched for related articles and found the quadratic variation can be calculated with $dF_t*dY_t$.[1]
Some common stochastic processes such as Brownian motion and Poisson processes are semimartingales.
[1] https://en.wikipedia.org/wiki/Quadratic_variation#It%C3%B4_processes:~:text=.-,Semimartingales,-%5Bedit%5D