Randomization in the first hitting time problem

Abstract

In this paper we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F (t), and a linear boundary, b(t) = µt, find a distribution of the initial state such that the distribution of the first hitting time is F (t). This problem has important applications in credit risk modeling where the process represents, so-called, distance to default of an obligor, the first hitting time represents a default event and the boundary separates the healthy states of the obligor from the default state. We show that randomization of the initial state of the process makes the problem analytically tractable.Primary Subjects: 60G40 Secondary Subjects: 91B70

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

“…Indeed, some care has to be used to exclude that the found distribution of η has atoms together with a density. However, as already noted in [2,7], the function g may not be the Laplace transform of some probability density function, so in that case the IFPT problem has no solution; really, it may admit more than one solution, since the right-hand member of Equation (4) essentially furnishes the moments of η of any order n, but this is not always sufficient to uniquely determine the density g of η. In line of principle, the right-hand member of Equation (4) can be expressed in terms of the Laplace transform of f (t) = F (t), though it is not always possible to do this explicitly.…”

“…This problem, also known as the generalized Shiryaev problem, was studied in [1,2,7,8], essentially in the case when X(t) is BM and S(t) is a straight line; note that the question of the existence of the solution is not a trivial matter (see e.g., [2,7]). In this paper, by using the properties of the exponential martingale, we extend the results to more general boundaries S.…”

“…The IFPT problem has interesting applications in mathematical finance , in particular in credit risk modeling, where the FPT represents a default event of an obligor (see [7]) and in diffusion models for neural activity ( [9]). 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]).…”

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

“…without reflecting (see e.g. [3]), [4], [25] ) the function g may not be the Laplace transform of some probability density function; in that case the IFPT problem has no solution. This is the reason why Theorem 2.4 is formulated in a conditional form.…”

“…[25]), in Biology, specially in the framework of diffusion models for neural activity (see e.g. [29]), and in Queueing theory (see e.g.…”