An inverse first-passage problem for one-dimensional diffusions with random starting point

On First-Hitting Time of a Linear Boundary by Perturbed Brownian Motion

“…By using the arguments of [2], with a replaced with , x a we are able to obtain the following results.…”

“…Then, for Y a x, the first-hitting time of S by XY (t) is Y = inf{t > 0 : XY (t) a + bt}. By proceeding in analogous manner as in [2], with a replaced by a x, and by correcting a typographical error, there present (see [4]), we obtain:…”

“…Indeed, this is possible when Z(t) can be reduced to Brownian motion by a deterministic transformation and a random time-change (see [2] for a few examples); then, the results concerning Z(t) can be obtained by those for B t , by using the arguments of [2] with a replaced by a x.…”

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.…”

“…Notice that, if we take a = 0 and b = 1, both boundaries are attainable and there is no need for reflection in a and b, because the process without reflecting cannot exit the interval [0, 1], for any time t (see e.g. [4]). If the FPT density is given by (3.6), from (3.7) we obtain that the solutions to the IFPT problems for the processes (i)-(iv) above, relative to the barrier S, are explicitly given by: g(x) = 3x 2/3 S 1/3 − a 1/3 −1 · 1 (a,S) (x) (i), g(x) = …”