First Passage Time Distributions for Brownian Motion with Drift and a Local Limit Theorem

2021 ◽  
pp. 191-198
Author(s):  
Rabi Bhattacharya ◽  
Edward C. Waymire
Keyword(s):  
Brownian Motion ◽  
Limit Theorem ◽  
First Passage Time ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Passage Time ◽  
First Passage ◽  
Brownian Motion With Drift

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 Three Essays on Stopping

Risks ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 105
Author(s):  
Eberhard Mayerhofer
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Diffusion Characteristic ◽  
Reflected Brownian Motion ◽  
First Passage ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
Closed Form Formula

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

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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