Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

Abundo, Mario

doi:10.1080/07362990600958804

Cited by 22 publications

(17 citation statements)

References 22 publications

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“…This kind of difficulty in showing the existence of a solution to an inverse first-passage time problem is common to another type of inverse problem (see e.g. [5]), in which one has to find the shape of the moving barrier S(t) in such a way that the FPT of X(t) over S(t) for deterministic fixed initial condition, has a predetermined distribution F.…”

Section: The Ifpt Problem For Reflected Brownian Motion With Driftmentioning

confidence: 99%

“…A numerical solution to that inverse problem was found in [5]; the existence of the solution is, at the moment, a still open problem (see e.g. [46]).…”

Section: The Ifpt Problem For Reflected Brownian Motion With Driftmentioning

confidence: 99%

“…The inverse FPT problem for diffusions generally focuses on determining the barrier S, when f (t|x) is given (see e.g. [5], [46]). Since we assume that the initial position η is random, we consider a slight modification of the problem, that is the following inverse first-passage-time (IFPT) problem.…”

Section: Introductionmentioning

confidence: 99%

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One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

Abundo

2014

Stochastic Analysis and Applications

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We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion X(t) reflected between two boundaries a and b, which starts from a random position η. Let a ≤ S ≤ b be a given threshold, such that P (η ∈ [a, S]) = 1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the first-hitting time of X to S has distribution F. This is a generalization of the analogous problem for ordinary diffusions, i.e. without reflecting, previously considered by the author.

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Section: The Ifpt Problem For Reflected Brownian Motion With Driftmentioning

confidence: 99%

“…A numerical solution to that inverse problem was found in [5]; the existence of the solution is, at the moment, a still open problem (see e.g. [46]).…”

Section: The Ifpt Problem For Reflected Brownian Motion With Driftmentioning

confidence: 99%

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One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

Abundo

2014

Stochastic Analysis and Applications

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“…S is not sufficient to determinate uniquely the desired solution of the PDDE (14), because it is a second order equation. We will return to this problem when we will consider some explicit examples.…”

Section: Thus By Taking Expectation One Obtains E(x(t)) = N(t) + E(jmentioning

confidence: 99%

Investigating the Distribution of the First-Crossing Area of a Diffusion Process with Jumps Over a Threshold

Abundo¹

2013

TOAMJ

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For a given barrier S and a one-dimensional jump-diffusion process ), (t X starting from , < S x we study the. S In particular, we show that the Laplace transform and the moments of ) (x A S are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum ofstudied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by ) (t X till its firstpassage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.

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“…Notice, however, that another type of inverse first-passage problem can be considered: it consists in determining the boundary shape S, when the FPT distribution F and the starting point η are assigned (see e.g., [10][11][12][13]). …”

Section: Introductionmentioning

confidence: 99%

The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

Abundo

2018

Mathematics

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Let X(t) be a continuously time-changed Brownian motion starting from a random position η, S(t) a given continuous, increasing boundary, with S(0) ≥ 0, P(η ≥ S(0)) = 1, and F an assigned distribution function. We study the inverse first-passage time problem for X(t), which consists in finding the distribution of η such that the first-passage time of X(t) below S(t) has distribution F, generalizing the results, valid in the case when S(t) is a straight line. Some explicit examples are reported.

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Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

Cited by 22 publications

References 22 publications

One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

Investigating the Distribution of the First-Crossing Area of a Diffusion Process with Jumps Over a Threshold

The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion

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