Optimal Stopping Problem For Processes With Independent Increments

We consider the optimal stopping problem for processes with independent increments with the exponential g(x) = (1 − e −x ) + or logarithmic g(x) = (ln x) + payoff function. For the exponential payoff function, it is shown that the optimal stopping time is the first time of hitting a certain level. For the logarithmic payoff function, it is proved that a moment of the first hitting of a level cannot be optimal.