On an integral equation for first-passage-time probability densities

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

A note on the evaluation of first-passage-time probability densities

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

Stochastic barriers for the Wiener process

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ T W(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ T W(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

Evaluations of absorption probabilities for the Wiener process on large intervals

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦T W(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

Boundaries with negative jumps for the Brownian motion

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.

On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Lett → h(t) be a smooth function on ℝ+, andB= {Bs;s≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variablesTh: = inf {S;Bs=h(s)} and λth: = sup {s≦t; Bs= h(s)}, wheret>0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

On the Computation of the Survival Probability of Brownian Motion with Drift in a Closed Time Interval When the Absorbing Boundary Is a Step Function

This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed.

A transition to sharp timing in stochastic leaky integrate-and-fire neurons driven by frozen noisy input

The firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current (Mainen & Sejnowski, 1995 ). This suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit. Modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking (Cecchi et al., 2000 ; Tiesinga, 2002 ; Fellous, Rudolph, Destexhe, & Sejnowski, 2003 ). In order to investigate this interplay, we focus on the stochastic leaky integrate-and-fire neuron and identify the Hölder exponent H of the integrated input as the key mathematical property dictating the regime of firing of a single-unit neuron. We have recently provided numerical evidence (Taillefumier & Magnasco, 2013 ) for the existence of a phase transition when [Formula: see text] becomes less than the statistical Hölder exponent associated with internal gaussian white noise (H=1/2). Here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent H>0. In doing so, we account for the existence of a transition between two regimes of firing when H=1/2, and we show that spiking times have a continuous density when the Hölder exponent satisfies H>1/2. The transition at H=1/2 formally separates rate codes, for which the neural firing probability varies smoothly, from temporal codes, for which the neuron fires at sharply defined times regardless of the intensity of internal noise.

Partial barrier-absorption probabilities for the Wiener process

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The techniques of computing probabilities of the type are well known. The main purpose of this paper is to present ways of finding barrier-absorption probabilities when the barrier function is defined only on sub-intervals of [0, T].