Why can't we neglect the $dt$ there?

$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$

  • $\begingroup$ Hi: If you ignore first order terms, then you won't have a taylor expansion anymore because there won't be anything left. The convention is to ignore second order terms which, in non-stochastic case, is $dx^2$. Unfortunately, in the stochastic framework of Ito, $dx^2 = dt$ so it can't be ignored. Note that this is a heuristicy argument. I would read something on the net of an introductory nature and hopefully get a more formal explanation. There are many documents out there so I don't want to pick one for you. Some are more theoretical than others but, either way, there are a ton. Good luck. $\endgroup$
    – mark leeds
    Commented Oct 28, 2020 at 19:14
  • $\begingroup$ Basically this extra term is the whole point of Ito calculus and the often first thing that is discussed in any textbook about the subject. It is there due to the non-vanishing of the quadratic variation for a Brownian motion. $\endgroup$ Commented Oct 30, 2020 at 6:01