About Ito's Integral

Question

Is such an integral an Ito Integral?

$$\int{B_t dB_t} $$

where $B_t$ is a Wiener Process. Shouldn't the integrand be of "bounded variation" ?

Answer 1

There are no requirements of bounded variation for the integrand in the Itô integral. The only requirements on the integrand are that it is an adapted process, measurable wrt the product measure (Lebesgue × probability measure), and an $L^2$ criterion.