I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing time)
"As in the statement" means that the $g_n$ are bounded on compact subsets of $R^+ \times R$, and the eqn 4.1.9 it refers to is the 1d ito formula...
I start by splitting $min(t,t_n)$ into two cases $$min(t,t_n)=t\rightarrow \int_0^{min(t,t_n)}v\frac{\partial g_n}{\partial x}dB_s=\int_0^{t}v\frac{\partial g_n}{\partial x}dB_s=\int_0^{t}v\frac{\partial g}{\partial x}dB_s=\int_0^{min(t,t_n)}v\frac{\partial g}{\partial x}dB_s$$ $$min(t,t_n)=t_n\rightarrow \int_0^{min(t,t_n)}v\frac{\partial g_n}{\partial x}dB_s=\int_0^{t_n}v\frac{\partial g_n}{\partial x}dB_s=\int_0^{t_n}v\frac{\partial g}{\partial x}dB_s=\int_0^{min(t,t_n)}v\frac{\partial g}{\partial x}dB_s$$
So , yes... $\int_0^{min(t,t_n)}v\frac{\partial g_n}{\partial x}dB_s=\int_0^{min(t,t_n)}v\frac{\partial g}{\partial x}dB_s$ (...?)
...But I'm pretty sure that it can not be that simple, I never used any reference to anything being bounded, so there is something that I may be missing. Probably on the "holds for any $n$ part)
Any help would be welcome on this, thanks !