With $B$ a standard Brownian motion, write $ dX_t=f_tdt+g_tdB_t. $ What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists?
I think $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ should be adapted to the filtration of $B$.
Do they have to be càdlag?
What about the integrability condition \begin{align} &E\left( \int_0^t f_sds\right) < \infty, \text{ and },\\ &E\left( \int_0^t g_s^2d[B]_s\right) < \infty? \end{align}
We are using several textbooks in our class, and I can't pindown the theorem that treats this.
(Kurtz, Stochastic Analysis; Oksendal; Protter)