scholarly journals First passage time statistics of Brownian motion with purely time dependent drift and diffusion

2011 ◽  
Vol 390 (11) ◽  
pp. 1841-1852 ◽  
Author(s):  
A. Molini ◽  
P. Talkner ◽  
G.G. Katul ◽  
A. Porporato
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
First Passage ◽  
And Diffusion

scholarly journals The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains

2021 ◽  
Vol 66 (1) ◽  
pp. 175-195
Author(s):  
JM Lee ◽  
JM Lee
Keyword(s):  
Boundary Condition ◽  
Brownian Motion ◽  
Heat Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
First Passage ◽  
Time Density

В книге [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849] доказано, что плотность момента первого пересечения границы одномерным стандартным броуновским движением будет непрерывной, когда граница непрерывна по Гeльдеру с показателем больше $1/2$. С целью распространить результат [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849] на многомерные области мы показываем, что существует непрерывная функция плотности момента первого пересечения подвижных границ в $\mathbf R^d$, $d \ge 2$, стандартным $d$-мерным броуновским движением при $C^3$-диффеоморфизме. Как и в [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849], используя свойство локального времени стандартного $d$-мерного броуновского движения и уравнение теплопроводности с граничным условием Дирихле, мы находим достаточное условие существования непрерывной функции плотности.

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 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.

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