scholarly journals The First-passage Time of the Brownian Motion to a Curved Boundary: an Algorithmic Approach

2016 ◽  
Vol 38 (1) ◽  
pp. A196-A215 ◽  
Author(s):  
S. Herrmann ◽  
E. Tanré
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Algorithmic Approach ◽  
Curved Boundary

scholarly journals On the First Passage time for Brownian Motion Subordinated by a Lévy Process

2009 ◽  
Vol 46 (1) ◽  
pp. 181-198 ◽  
Author(s):  
T. R. Hurd ◽  
A. Kuznetsov
Keyword(s):  
Brownian Motion ◽  
Approximation Scheme ◽  
First Passage Time ◽  
Gaussian Model ◽  
Passage Time ◽  
Financial Mathematics ◽  
First Passage ◽  
Financial Modeling ◽  
Motion Time ◽  
Normal Inverse Gaussian

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) 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 the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the 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 led 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.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (02) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

scholarly journals On Durbin's Series for the Density of First Passage Times

2011 ◽  
Vol 48 (03) ◽  
pp. 713-722
Author(s):  
P. Zipkin
Keyword(s):  
Pointwise Convergence ◽  
First Passage Time ◽  
Convergent Series ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Substitution Method ◽  
Curved Boundary ◽  
Weiner Process ◽  
Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (2) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

scholarly journals The inverse first passage time problem for killed Brownian motion

2020 ◽  
Vol 30 (3) ◽  
pp. 1251-1275
Author(s):  
Boris Ettinger ◽  
Alexandru Hening ◽  
Tak Kwong Wong
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Time Problem ◽  
First Passage Time Problem
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