Given a Brownian movement $ {W_t } _ {t in[0;T]$ and a continuous process, adapted and square-integrable (bounded if desired) $ { sigma_t } _ {t in[0;T]$ Y $ varepsilon> 0 $, I want to prove that there is a $ delta> 0 $ such that

For all $ s in [0;T]$ and all $ M en mathcal F_s $, is

$$

mathbb E bigg (1_M max_ {s le t le (s + delta) wedge T} bigg | int_s ^ t sigma_u mathrm dW_u bigg | bigg)

le varepsilon.

$$

by $ sigma equiv 1 $, this is easy because we only consider

$$

mathbb E Big (1_M max_ {s le t le (s + delta) wedge T} | W_t – W_s | Big)

$$

for which we have a limit due to the distribution of the maximum $ W $ and the increases in Brownian motion are independent of the past.

Is there anything similar for arbitrary Ito integrals (or those that satisfy some assumptions)?