Consider the stochastic differential equation:$$dx(t)=f(x(t))dt+g(x(t))dB(t)$$ where $B(t)$ is the standard brownian motion.
Is the following always true:
$$\mathbb{E}\int_0^tH(x(t))dB(t)=0$$
If it is not true in general, what are the conditions on the function $H(x)$ that makes the equality holds?
If the stochastic integral wrt to the B.M. is well defined, then your claim is always true. For the integral to be well defined we need the integrand process H to be $\{\mathscr{F}_t\}$-adapted and also to satisfy $$\int_0^t |H_u|^2 du < \infty.$$