Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes

The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is a long-standing problem with no known closed-form solution except in specific cases. Taking this as a starting-point, and extending to a general mean-reverting process, we investigate the long- and short-time asymptotics using a combination of Hopf-Cole and Laplace transform techniques. As a result we are able to give a single formula that is correct in both limits, as well as being exact in certain special cases. We demonstrate the results using a variety of other models

Collective Decision-Making in Ideal Networks: The Speed-Accuracy Tradeoff

We study collective decision-making in a model of human groups, with network interactions, performing two alternative choice tasks. We focus on the speed-accuracy tradeoff, i.e., the tradeoff between a quick decision and a reliable decision, for individuals in the network. We model the evidence aggregation process across the network using a coupled drift diffusion model (DDM) and consider the free response paradigm in which individuals take their time to make the decision. We develop reduced DDMs as decoupled approximations to the coupled DDM and characterize their efficiency. We determine high probability bounds on the error rate and the expected decision time for the reduced DDM. We show the effect of the decision-maker's location in the network on their decision-making performance under several threshold selection criteria. Finally, we extend the coupled DDM to the coupled Ornstein-Uhlenbeck model for decision-making in two alternative choice tasks with recency effects, and to the coupled race model for decision-making in multiple alternative choice tasks.Comment: to appear in IEEE TCN

Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a parameter translation, this result can be extended to an inhomogeneous OU-process. Next, we derive a general formula for the joint distribution and the survival functions for the maxima of a continuous Markov process in consecutive periods. With these results, one can obtain semi-analytical expressions for the joint distribution and the multivariate survival functions for the maxima of an OU-process, with piecewise constant parameter functions, in consecutive time periods. The joint distribution and the survival functions can be evaluated numerically by an iterated quadrature scheme, which can be implemented efficiently by matrix multiplications. Moreover, we show that the computation can be further simplified to the product of single quadratures if the filtration is enlarged. Such results may be used for the modelling of heatwaves and related risk management challenges.Comment: 38 pages, 10 figures, 2 table

First Passage Time Densities in Non-Markovian Models with Subthreshold Oscillations

Motivated by the dynamics of resonant neurons we consider a differentiable, non-Markovian random process $x(t)$ and particularly the time after which it will reach a certain level $x_b$. The probability density of this first passage time is expressed as infinite series of integrals over joint probability densities of $x$ and its velocity $\dot{x}$. Approximating higher order terms of this series through the lower order ones leads to closed expressions in the cases of vanishing and moderate correlations between subsequent crossings of $x_b$. For a linear oscillator driven by white or coloured Gaussian noise, which models a resonant neuron, we show that these approximations reproduce the complex structures of the first passage time densities characteristic for the underdamped dynamics, where Markovian approximations (giving monotonous first passage time distribution) fail