The first-passage area of Ornstein-Uhlenbeck process revisited

For Ornstein-Uhlenbeck process X(t), starting from X(0) = x > 0, we highlight some results about the first-passage time of X(t) through zero and its first-passage area, that is the random area swept out by X(t), till its first-passage through zero. We study single and joint moments of the first-passage time and first-passage area, and their behaviors, as x -> 0(+) and x -> +infinity; moreover, we investigate the expected value of the time average of X(t) till the FPT, and the maximum displacement of X(t).