Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process

Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density. 1 Research supported by RiskLab, Switzerland, funded by Credit Suisse Group, Swiss Re and UBS AG. The third author was supported by a Steno grant from the Danish Natural Science Research Council.