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On the Two-Boundary First-Crossing-Time Problem for Diffusion Processes

Authors:
Aniello buonocore Buonocore at University of Naples "L'Orientale"
Aniello buonocore Buonocore
Virginia Giorno at Università degli Studi di Salerno
Virginia Giorno
Amelia GIUSEPPINA Nobile at Università degli Studi di Salerno
Amelia GIUSEPPINA Nobile
L. M. Ricciardi
L. M. Ricciardi
L. M. Ricciardi
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Abstract

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.’s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.
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On the Two-Boundary First-Crossing-Time Problem for Diffusion Processes
A. Buonocore; V. Giorno; A. G. Nobile; L. M. Ricciardi
Journal of Applied Probability, Vol. 27, No. 1. (Mar., 1990), pp. 102-114.
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Mon Jan 15 06:24:42 2007
... The integral-equation method was pioneered by Durbin (1971), who used it to compute the power of the Kolmogorov-Smirnov test. The method was subsequently developed by Ricciardi and colleagues (Buonocore et al., 1987;Buonocore, Giorno, Nobile, & Ricciardi, 1990;Ricciardi, 1976;Ricciardi & Sato, 1983) to characterize the firing time distributions of model integrate-and-fire neurons. Early applications to decision processes were described by Heath (1992) and Smith (1995Smith ( , 1998. ...
... Equation A1 is a Volterra integral equation of the first kind, in which the unknown function of interest, the first-passage time density function g[a(t), t | z, 0], appears under the integral sign. For processes with two absorbing boundaries, as often arise in modeling decision processes, there are two such equations, one for each boundary, and the decomposition based on first boundary crossings needs to keep track of crossings at each of them separately (Buonocore et al., 1990;Smith, 2000;Smith & Ratcliff, 2022), but I restrict myself here to the single boundary case, which is the one that arises in models with racing processes. Buonocore, Nobile, and Ricciardi (1987) showed that it is possible to transform Eq. ...
... by trapezoidal integration. The discretized form of the equation (Buonocore et al., 1990;Smith, 2000, pp. 440-441) Buonocore et al. (1987) showed that, when the kernel is chosen in the manner they prescribed, the discretized firstpassage time density obtained from Eq. A3 converges to the true first-passage time density as becomes small. ...
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... If the two boundaries are absorbing, following ref. [78], one can introduce P ...
... (N |N in , z in ) and P (0) + (N |N in , z in ) satisfy the two coupled integral equations[78] ...
... are obtained from first principles in ref.[78] and we do not reproduce their derivation here. It is nonetheless worth pointing out that they can be generalised to any drift function and noise amplitude in the Langevin equation, although, for practical use, one needs to compute the solution f of the Fokker-Planck equation in the absence of boundaries [as was done in eq. ...
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... Buonocore, Nobile, and Ricciardi (1987) reformulated the equation to a Volterra integral equation of second kind to remove the singularities. In a second work, the two-thresholds case has been considered (Buonocore, Giorno, Nobile, & Ricciardi, 1990). For a survey and an overview on the derivations and reformulations into the Volterra equation of second kind, we refer to Smith (2000). ...
... By KFE (transformed) we denote the modification of the KFE solution described in the present work. The integral equation method is the one introduced in Buonocore et al. (1990). Blue labels indicate the technique that is used to show the relation between different models, black arrows refer to the discretization technique. ...
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... Using the method of Volterra integral equations as in Ref. [70], one can show that γ AE ðNjϕ 0 ; N 0 Þ satisfy the following integral relations (see Appendix A for further details): ...
... H. F. acknowledges partial support from the "Saramadan" federation of Iran. A. N. thanks S. Hooshangi for helpful discussions. In this section, we follow the same process used by Ref. [70] to obtain a similar set of equations for γ AE ðτÞ in the case where one of the boundaries has Brownian motion. We denote the two boundaries by S 1 and S 2 , respectively. ...
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