I'm working through Protter's book on stochastic integration; this is problem 16 from chapter 2. I can't seem to crack it--maybe someone here can give me a hint? Let B be standard Brownian and H be a continuous, adapted process. I'm trying to prove that $$ \lim_{h \to 0} \frac{1}{B_{t+h} - B_t}\int_t^{t+h} H_sdB_s = H_t, $$ where the limit is taken in probability.
This looks a lot like the classical mean-value theorem, but I can't see how to apply that theorem or emulate its proof. Ito's formula lets us rewrite the problem as $$ \lim_{h \to 0} \frac{1}{B_{t+h} - B_t}\int_t^{t+h} (H_s - H_t)dB_s = 0. $$ But I'm not sure that helps. The integral goes to zero: continuity of $H$ upgrades to ucp continuity on $[t,t+h]$ (its a compact set) and the stochastic integral preserves ucp. But the denominator $B_{t+h} - B_t$ also goes to zero. In a classical setting I'd try l'Hopital, but nothing here is differentiable.
Also, it's interesting to me that the limit is in probability--not ucp. That suggests to me that commuting limits is not the right approach. Do I want to be looking at the variance of this expression (show that it goes to zero)? It's not obvious to me how to do that either...
Any observations would be greatly appreciated!