On the first passage time for Brownian motion subordinated by a Levy process

TL;DR

This paper investigates the relationship between standard and second-kind first passage times for Levy processes, proposing an iterative approximation method that enhances numerical computation efficiency, especially for models like Variance Gamma.

Contribution

It establishes that the standard first passage time is the almost sure limit of iterations of the second-kind passage time and develops fast algorithms for specific models.

Findings

The iterative scheme converges to the standard first passage time.

The method is competitive with existing numerical techniques.

For the Variance Gamma model, the algorithm is particularly fast.

Abstract

This paper considers the class of L\'evy processes that can be written as a Brownian motion time changed by an independent L\'evy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are lead to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.