I have a question concerning a calculus "trick" sometimes used in stochastic calculus (e.g. in the Book on Arbitrage Theory in Cont. Time of Bjoerk). There they do the following in the proof of Prop. 29.6:
$\frac{1}{B_t}dF_t=h_tdW_t$, multiply on both sides with $B_t$ to obtain $dF_t=B_th_tdW_t$. The step seems rather obvious. $B_t$ is a prob. a finite variation process. The question is: Can one apply this also in general?
I.e. $F$ is a solution to
$dM_t = F_td\frac{1}{S_t}$ iff it solves $S_tdM_t = S_tF_t \frac{1}{S_t}$
$M$ depends on F, but is a local martingale. Say $F_td\frac{1}{S_t}$ is a (strict) local martingale. Multiplying $S$ gives $S_t F_td\frac{1}{S_t}$ which might well be a true martingale. So this transformation can have significant impact. Is it allowed?
Thank you for you attention!