# What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero:

$$E\left[\int_0^tf(W_s,s)dW_s\right]=0$$

Any reference?

ps: it is clear that if $f(\cdot)$ is bounded, it will be zero, but I am pretty sure there must be weaker conditions.

The requirement is simply that $f$ is integrable. An ito integral is approximated by $\sum_{s_i} f(W_{s_i}, s_i) (W_{s_{i+1}} - W_{s_i})$. Since $f(W_{s_i}, s_i)$ is independent of that interval (by definition of brownian motion), the expectation of each summand is zero. The result follows.