Asymptotics of hitting probabilities for general one-dimensional pinned diffusions

Abstract

We consider a general one-dimensional diffusion process and we study the probability of crossing a boundary for the associated pinned diffusion as the time at which the conditioning takes place goes to zero. We provide asymptotics for this probability as well as a first order development. We consider also the cases of two boundaries possibly depending on the time. We give applications to simulation.

Simulation of stopped diffusions

“…Remark 1. In the context of approximating killed diffusions, it was pointed out that this frozen coefficient approximation gives incorrect asymptotics and that more sophisticated approximations should be used [18,19]. The application of these ideas to stopped diffusions remains a topic of ongoing research.…”

Binomial approximation of Brownian motion and its maximum

“…In Walsh (2003), the author works with exponential binomial trees and, by embedding techniques, he obtains an expansion of the error; this is a really stronger result than ours, but it holds for functions which are at least piecewise C 2 : For similar convergence problems involving the maximum process, one can also see Baldi (1995), Baldi and Caramellino (2002), Gobet (2000) (all connected with the barrier case), and Tonou (1997), where there is only a partial proof, but for a more general context.…”

Bounds for the transition density of time-homogeneous diffusion processes

“…The proof is a straight-forward application of Girsanov's theorem and its idea is similar to the one used in Baldi and Caramellino (2002). We present the proof for case [A], the proof for case [B] is completed similarly (see Downes and Borovkov (2008) for the general approach).…”

“…We use Girsanov's theorem and then a transformation of the Radon-Nikodym density of the type suggested in Baldi and Caramellino (2002) to relate probabilities for a general diffusion (U t ) to those of a 'reference diffusion'. Using a reference diffusion with known transition functions, we are able to derive various bounds for the transition functions under mild conditions on the original process.…”