On hitting times for compound Poisson dams with exponential jumps and linear release rate

2001 ◽  
Vol 38 (03) ◽  
pp. 781-786 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Release Rate ◽  
Shot Noise ◽  
Hypergeometric Functions ◽  
Hitting Times ◽  
Stieltjes Transform ◽  
Compound Poisson ◽  
Confluent Hypergeometric Functions ◽  
Kummer Functions ◽  
Shot Noise Process ◽  
The Mean

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

On hitting times for compound Poisson dams with exponential jumps and linear release rate

2001 ◽  
Vol 38 (3) ◽  
pp. 781-786 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Release Rate ◽  
Shot Noise ◽  
Hypergeometric Functions ◽  
Hitting Times ◽  
Stieltjes Transform ◽  
Compound Poisson ◽  
Confluent Hypergeometric Functions ◽  
Kummer Functions ◽  
Shot Noise Process ◽  
The Mean

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

scholarly journals A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

2012 ◽  
Vol 49 (01) ◽  
pp. 100-113 ◽  
Author(s):  
Hermine Biermé ◽  
Agnès Desolneux
Keyword(s):  
Shot Noise ◽  
Poisson Point Process ◽  
Functions Of Bounded Variation ◽  
Level Crossing ◽  
Level Crossings ◽  
Variable Formula ◽  
Shot Noise Process ◽  
Change Of Variable ◽  
The Mean ◽  
The Fourier Transform

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (01) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

scholarly journals A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

2012 ◽  
Vol 49 (1) ◽  
pp. 100-113 ◽  
Author(s):  
Hermine Biermé ◽  
Agnès Desolneux
Keyword(s):  
Shot Noise ◽  
Poisson Point Process ◽  
Functions Of Bounded Variation ◽  
Level Crossing ◽  
Level Crossings ◽  
Variable Formula ◽  
Shot Noise Process ◽  
Change Of Variable ◽  
The Mean ◽  
The Fourier Transform

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

The cost of a general stochastic epidemic

1972 ◽  
Vol 9 (02) ◽  
pp. 257-269 ◽  
Author(s):  
J. Gani ◽  
D. Jerwood
Keyword(s):  
Recursive Equation ◽  
Stieltjes Transform ◽  
Stochastic Epidemic ◽  
The Mean ◽  
The Cost ◽  
Mean Cost ◽  
General Stochastic

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.

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