35 research outputs found
Analysis and numerical solution of an inverse first passage problem from risk management
We study the following "inverse first passage time" problem. Given a diffusion process Xt and aprobability distribution q(t) on t &ge 0, does there exist a boundary b(t) such that q(t)=P[&tau &ge t], where &tau is the first hitting time of Xt to the time dependent level b(t). We formulate the inverse first passage time probleminto a free boundary problem for a parabolic partial differential operator and prove there exists a unique viscosity solution to the associated Partial Differential Equation by using the classical penalization technique. In order to compute the free boundary with a given default probability distribution, we investigate the small time behavior of the boundary b(t), presenting both upper and lower bounds first. Then we derive some integral equations characterizing the boundary. Finally we apply Newton-iteration on one of them to compute the boundary. Also we compare our numerical scheme with some other existing ones
One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem
We study an inverse first-hitting problem for a one-dimensional,
time-homogeneous diffusion reflected between two boundaries and
which starts from a random position Let be a given
threshold, such that and an assigned distribution
function. The problem consists of finding the distribution of such that
the first-hitting time of to has distribution This is a
generalization of the analogous problem for ordinary diffusions, i.e. without
reflecting, previously considered by the author
The arctangent law for a certain random time related to a one-dimensional diffusion
For a time-homogeneous, one-dimensional diffusion process we
investigate the distribution of the first instant, after a given time at
which exceeds its maximum on the interval generalizing a result
of Papanicolaou, which is valid for Brownian motion
On the excursions of drifted Brownian motion and the successive passage times of Brownian motion
By using the law of the excursions of Brownian motion with drift, we find the
distribution of the th passage time of Brownian motion through a straight
line In the special case when we extend the result to
a space-time transformation of Brownian motion.Comment: 4 figures, accepted for publication in Physica