The behavior near the origin of the supremum functional in a process with stationary independent increments

1975 ◽  
Vol 12 (01) ◽  
pp. 159-160
Author(s):  
Sidney I. Resnick ◽  
Michael Rubinovitch
Keyword(s):  
Lévy Measure ◽  
Independent Increments ◽  
Positive Jump

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z 1(t) = X(t), Z 2(t) = sup0≦s≦t X(s) and Z 3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: lim t↓0 t —1 P[Zi (t) > x] = M +(x) at all points of continuity of M +, the Lévy measure of X.

The behavior near the origin of the supremum functional in a process with stationary independent increments

1975 ◽  
Vol 12 (1) ◽  
pp. 159-160
Author(s):  
Sidney I. Resnick ◽  
Michael Rubinovitch
Keyword(s):  
Lévy Measure ◽  
Independent Increments ◽  
Positive Jump

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (04) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pk f of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tk f of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Remarks on the absolute maximum of a Lévy process

2002 ◽  
Vol 39 (02) ◽  
pp. 282-295
Author(s):  
Mykola Bratiychuk
Keyword(s):  
Asymptotic Behaviour ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Absolute Maximum ◽  
Lévy Measure ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Absolute

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

Remarks on the absolute maximum of a Lévy process

2002 ◽  
Vol 39 (2) ◽  
pp. 282-295
Author(s):  
Mykola Bratiychuk
Keyword(s):  
Asymptotic Behaviour ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Absolute Maximum ◽  
Lévy Measure ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Absolute

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (4) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

2021 ◽  
Vol 58 (1) ◽  
pp. 42-67 ◽  
Author(s):  
Mads Stehr ◽  
Anders Rønn-Nielsen
Keyword(s):  
Space Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Time Model ◽  
Spatial Object ◽  
Tail Behaviour ◽  
Fixed Radius ◽  
The Right ◽  
Space Time Model

AbstractWe consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

scholarly journals Global sensitivity analysis for stochastic processes with independent increments

2020 ◽  
Vol 62 ◽  
pp. 103098
Author(s):  
Emeline Gayrard ◽  
Cédric Chauvière ◽  
Hacène Djellout ◽  
Pierre Bonnet ◽  
Don-Pierre Zappa
Keyword(s):  
Sensitivity Analysis ◽  
Stochastic Processes ◽  
Global Sensitivity Analysis ◽  
Global Sensitivity ◽  
Independent Increments ◽  
Processes With Independent Increments

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

2004 ◽  
Vol 07 (01) ◽  
pp. 131-145 ◽  
Author(s):  
UWE FRANZ ◽  
NICOLAS PRIVAULT
Keyword(s):  
Lie Algebras ◽  
Lévy Processes ◽  
Unitary Operator ◽  
Levy Processes ◽  
Absolutely Continuous ◽  
Commutative Subalgebra ◽  
Independent Increments ◽  
Commutative Subalgebras ◽  
Current Representation ◽  
General Method

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

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