The one-sided barrier problem for an integrated ornstein-uhlenbeck process

From Poisson shot noise to the integrated Ornstein–Uhlenbeck process: Neurally principled models of information accumulation in decision-making and response time

“…It is not clear why the two-barrier model fits better than the one-barrier model. It may be because the Hesse (1991) solution is only an approximation, which overestimates the mass in the distribution tail slightly under some conditions (see Hesse, 1991, Figures 1-4).…”

“…However, one useful result was obtained by Hesse (1991), who derived an analytic approximation to the first-passage-time density for an integrated OU process through a single absorbing barrier. Although this is a simpler problem than the one we have been considering here, it is nevertheless interesting, because it provides a way to study the properties of the integrated OU process when a barrier is present.…”

“…A more neurally plausible value for the decay coefficient would therefore be around α = 50. With such a value of α, the effects of the Poisson shocks decay to around 10% of their initial value in under 50 ms. Because the Hesse (1991) approximation becomes numerically unstable with such large values of α, I investigated a two-barrier process by Monte Carlo simulation.…”

The Poisson shot noise model of visual short-term memory and choice response time: Normalized coding by neural population size

“…Unlike the Wiener process, there are few closed-form, first-passage-time results known for the integrated OU process. Smith (2000) investigated the first-passage-time properties of a single-barrier process using a rational approximation derived by Hesse (1991) and of a two-barrier process using Monte-Carlo simulation. For barriers of both kinds, he showed that families of RT distributions for an integrated OU process were well approximated by a Wiener diffusion model.…”

First-Passage Distributions of Bidimensional Processes

“…An important special case of the system defined by Eqs. A number of papers have been published on first-passage problems for integrated processes (see, e.g., Hesse [5], Lachal [8], Lefebvre [9,10]). Then, under appropriate smoothness conditions on the functions / a n d a (see, e.g., Arnold [2, p. 152]), the j>(0 process is a diffusion process.…”