First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

Details First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

2007-08-27

arxiv.org/abs/0708.3562v1

Mathematics

Probability

27 pages, 3 figures

Scientific paper

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to whole real line this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.

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