Consider the Ito stochastic process $$X_t = X_0 + \int_{0}^{t} a_s ds + \int_{0}^{t} b_s dW_s$$
What conditions are necessary or sufficient (besides adaptability/measurability) to show that $$ E \left\lbrace \int_{0}^{t} b_s dW_s \right\rbrace = 0?$$
The main one I see is $$ E \left\lbrace \int_{0}^{t} b_s^2 ds \right\rbrace < \infty $$.
Are there any others that are sufficient or necessary? If the expectation isn't finite, can it still be true? I've only recently begun to learn about Ito integrals in more depth.
Let $Y_t=\int_0^t b_s dW_s$ in the Ito sense. If $E(|Y_t|)<\infty$, then $Y_t$ forms a martingale. Martingales have constant expectation, which must be zero, since $Y_0=0$ a.s.