A Property of the Ito Integral

Question

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then

$E[\int^{T}_{S}f dB_t]=0$

Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking limits. Intuitively to me this property does not seem true and I cannot see how to prove it exactly.

Answer 1

The proof is in two steps:

• it is true for elementary functions: you just have to prove that for every $Y_i$ measurable with respect to $F_{t_{i-1}}$, $$ E\left[\sum_{i=0}^{N-1} Y_i (B_{t_i} - B_{t_{i-1}})\right] =0 $$(it is quite easy).
• then, write an element $f$ as an $L^2$ limit of such elementary processes $f_n$. Then, $$ E \left[\int_S^T f_n dB - \int_S^T f dB \right]^2 = E\int_S^T |f_n - f|^2 ds\to 0 $$ via Ito's isometry.