Suppose $X=(X_t)$ is a stochastic process. I have a question about the notation $X_0$. Am I right, that $X_0(\omega)$ is not a constant. It depends on $\omega$ and can therefore have different values for different $\omega$? When do we know that $X_0$ is always a constant? Are there assumptions on the filtration? Suppose I know that $M_t:=X_t-X_0$ is a local martingale. Can I conclude that $X_t$ is a local martingale too? Obviously if $X_0$ is a constant, this is true. This is the motivation for my question.
Thank you for your help