Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

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Abstract
The asymptotic behaviour of the first-passage-time pdf through a constant boundary for an Ornstein-Uhlenbeck process is investigated for large boundaries. It is shown that an exponential pdf arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time pdf. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary