scholarly journals Approximating the First Crossing-Time Density for a Curved Boundary

Bernoulli ◽  
1996 ◽  
Vol 2 (2) ◽  
pp. 133 ◽  
Author(s):  
Henry E. Daniels
Keyword(s):  
Curved Boundary ◽  
Crossing Time ◽  
First Crossing Time ◽  
Time Density

scholarly journals Compound Poisson Process with a Poisson Subordinator

2015 ◽  
Vol 52 (2) ◽  
pp. 360-374 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci ◽  
Shelemyahu Zacks
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Bell Polynomials ◽  
Compound Poisson ◽  
Convergence Results ◽  
Special Cases ◽  
Time Problem ◽  
Crossing Time ◽  
First Crossing Time ◽  
Time Density

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions.

scholarly journals Compound Poisson Process with a Poisson Subordinator

2015 ◽  
Vol 52 (02) ◽  
pp. 360-374 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci ◽  
Shelemyahu Zacks
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Bell Polynomials ◽  
Compound Poisson ◽  
Convergence Results ◽  
Special Cases ◽  
Time Problem ◽  
Crossing Time ◽  
First Crossing Time ◽  
Time Density

A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions.

scholarly journals First-crossing and ballot-type results for some nonstationary sequences

2007 ◽  
Vol 39 (02) ◽  
pp. 492-509 ◽  
Author(s):  
Claude Lefèvre
Keyword(s):  
Time Distribution ◽  
Numerical Evaluation ◽  
Random Variables ◽  
Parametric Family ◽  
Partial Sums ◽  
Variable Parameters ◽  
Type Formula ◽  
Polynomial Structure ◽  
Crossing Time ◽  
First Crossing Time

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

On the two-boundary first-crossing-time problem for diffusion processes

1990 ◽  
Vol 27 (01) ◽  
pp. 102-114 ◽  
Author(s):  
A. Buonocore ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equations ◽  
Diffusion Processes ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Time Problem ◽  
Crossing Time ◽  
Single Integral ◽  
First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

scholarly journals Kendall's identity for the first crossing time revisited

2001 ◽  
Vol 6 (0) ◽  
pp. 91-94 ◽  
Author(s):  
Konstantin Borovkov ◽  
Zaeem Burq
Keyword(s):  
Crossing Time ◽  
First Crossing Time

On the two-boundary first-crossing-time problem for diffusion processes

1990 ◽  
Vol 27 (1) ◽  
pp. 102-114 ◽  
Author(s):  
A. Buonocore ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equations ◽  
Diffusion Processes ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Time Problem ◽  
Crossing Time ◽  
Single Integral ◽  
First Crossing Time

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

scholarly journals First-crossing and ballot-type results for some nonstationary sequences

2007 ◽  
Vol 39 (2) ◽  
pp. 492-509 ◽  
Author(s):  
Claude Lefèvre
Keyword(s):  
Time Distribution ◽  
Numerical Evaluation ◽  
Random Variables ◽  
Parametric Family ◽  
Partial Sums ◽  
Variable Parameters ◽  
Type Formula ◽  
Polynomial Structure ◽  
Crossing Time ◽  
First Crossing Time

In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

1995 ◽  
Vol 32 (2) ◽  
pp. 316-336 ◽  
Author(s):  
A. G. Di Crescenzo ◽  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
Linear Process ◽  
Two Dimensional ◽  
Constructive Approach ◽  
One Dimensional ◽  
Absorbing Boundaries ◽  
Dimensional Diffusion ◽  
Crossing Time ◽  
Probability Densities ◽  
First Crossing Time

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

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