I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process:
$X_t = X_0 + \int_{0}^{t} a_s ds + \int_0^t b_s dW_s$
Is this kind of integral a martingale or local martingale. Is this connected with the integral $\int_{0}^{t} a_s ds$? (if it is present, the process is a martigale and in the opposite situation it is a local martingale?). What would happen if we have some proces $Y_t$ and our $X_t$ looks like that:
$X_t = X_0 + \int_{0}^{t} f(Y_s) ds + \int_0^t f(Y_s) dW_s$
I would love to read something more about those properties. I will appreciate any link/paper related to this topic.