EXPONENTIAL TRENDS OF ORNSTEIN-UHLENBECK

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein-Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. 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 p.d.f. 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.

[1]  A. Siegert.  On the First Passage Time Probability Problem , 1951 .

[2]  H. E. Daniels.  The minimum of a stationary Markov process superimposed on a U-shaped trend , 1969 .

[3]  Donald H. Perkel,et al.  Solutions for a stochastic model of neuronal spike production , 1970 .

[4]  J. Durbin.  Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test , 1971, Journal of Applied Probability.

[5]  R. Anderssen,et al.  On the numerical solution of Brownian motion processes , 1973, Journal of Applied Probability.

[6]  Shunsuke Sato,et al.  Evaluation of the first-passage time probability to a square root boundary for the Wiener process , 1977, Journal of Applied Probability.

[7]  Shunsuke Sato,et al.  On the moments of the firing interval of the diffusion approximated model neuron , 1978 .

[8]  L. Favella,et al.  On a weakly singular Volterra integral equation , 1981 .

[9]  Laura Sacerdote,et al.  MEAN VARIANCE AND SKEWNESS OF THE FIRST PASSAGE TIME FOR THE ORNSTEIN-UHLENBECK PROCESS , 1981 .

[10]  L. Ricciardi,et al.  FIRST PASSAGE TIME PROBLEMS AND SOME RELATED COMPUTATIONAL METHODS , 1982 .

[11]  Shunsuke Sato,et al.  Diffusion approximation and first passage time problem for a model neuron. II. Outline of a computation method , 1983 .

[12]  Shunsuke Sato,et al.  On an integral equation for first-passage-time probability densities , 1984, Journal of Applied Probability.

[13]  L. Ricciardi,et al.  Evaluation of first passage time densities for diffusion processes , 1985 .