Ultimate ruin probability ($\psi (u)=\mathbb{P}({\inf }_{t\ge 0}{U}_t<0)$)

The probability that the insurance surplus in a classical risk model becomes negative at some finite time, starting from capital $u\ge 0$. Ultimate survival has probability $1-\psi (u)$. A finite-horizon ruin probability instead restricts the crossing time to a specified time interval.