Weighted Occupation Time for Branching Particle Systems and a Representation for the Supercritical Superprocess

1994 ◽  
Vol 37 (2) ◽  
pp. 187-196 ◽  
Author(s):  
Steven N. Evans ◽  
Neil O'Connell
Keyword(s):  
Particle System ◽  
The State ◽  
Particle Systems ◽  
Occupation Time ◽  
Occupation Times ◽  
Branching Particle System

AbstractWe obtain a representation for the supercritical Dawson-Watanabe superprocessin terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems.

scholarly journals OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE EQUILIBRIUM BRANCHING SYSTEMS

2009 ◽  
Vol 12 (04) ◽  
pp. 593-612 ◽  
Author(s):  
PIOTR MIŁOŚ
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Domain Of Attraction ◽  
Interesting Case ◽  
Occupation Time ◽  
Infinite Variance ◽  
Dependence Structure ◽  
Branching Particle System ◽  
Stable Motion ◽  
Fractional Stable Motion

We establish limit theorems for the fluctuations of the rescaled occupation time of a (d, α, β)-branching particle system. It consists of particles moving according to a symmetric α-stable motion in ℝd. The branching law is in the domain of attraction of a (1 + β)-stable law and the initial condition is the equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate α/β < d < (1 + β)α/β, critical d = (1 + β)α/β and large d > (1 + β)α/β dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments.

scholarly journals OCCUPATION TIMES OF SUBCRITICAL BRANCHING IMMIGRATION SYSTEMS WITH MARKOV MOTION, CLT AND DEVIATION PRINCIPLES

2012 ◽  
Vol 15 (01) ◽  
pp. 1250002 ◽  
Author(s):  
PIOTR MIŁOŚ
Keyword(s):  
Large Time ◽  
Particle System ◽  
General Setting ◽  
Moderate Deviation ◽  
Occupation Times ◽  
Branching Particle System ◽  
Constant Rate ◽  
Branching Law ◽  
Functional Central Limit ◽  
Occupation Time Process

In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in ℝd and undergoing subcritical branching with a constant rate of V > 0. New particles immigrate to the system according to a homogeneous space-time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations under mild assumptions on the Markov family. In the general setting a functional central limit theorem as well as large and moderate deviation principles are proved. The subcriticality of the branching law determines the behaviour in large time scales and it "overwhelms" the properties of the particles' motion. For this reason the results are the same for all dimensions and can be obtained for a wide class of Markov processes (both properties are unusual for systems with critical branching).

Immigration processes associated with branching particle systems

1998 ◽  
Vol 30 (03) ◽  
pp. 657-675 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Particle Systems ◽  
Poisson Random Measure ◽  
General Representation ◽  
Branching Particle System ◽  
Convolution Semigroups ◽  
Entrance Law ◽  
Skew Convolution Semigroup ◽  
Special Case

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

2007 ◽  
Vol 10 (03) ◽  
pp. 439-464 ◽  
Author(s):  
ANNA TALARCZYK
Keyword(s):  
Local Time ◽  
Particle System ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Particle Systems ◽  
The Self ◽  
Intersection Local Time ◽  
Branching Particle System ◽  
Independent Particle ◽  
Density Process

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.

Immigration processes associated with branching particle systems

1998 ◽  
Vol 30 (3) ◽  
pp. 657-675 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Particle System ◽  
Random Measure ◽  
Particle Systems ◽  
Poisson Random Measure ◽  
General Representation ◽  
Branching Particle System ◽  
Convolution Semigroups ◽  
Entrance Law ◽  
Skew Convolution Semigroup ◽  
Special Case

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

THE FEYNMAN–KAC FORMULA AND PRICING OCCUPATION TIME DERIVATIVES

1999 ◽  
Vol 02 (02) ◽  
pp. 153-178 ◽  
Author(s):  
JULIEN-N. HUGONNIER
Keyword(s):  
Brownian Motion ◽  
Time Derivative ◽  
Occupation Time ◽  
Barrier Options ◽  
Occupation Times

In this paper, we undertake a study of occupation time derivatives that is derivatives for which the pay-off is contingent on both the terminal asset's price and one of its occupation times. To this end we use a formula of M. Kac to compute the joint law of Brownian motion and one of its occupation times. General pricing formulas for occupation time derivatives are established and it is shown that any occupation time derivative can be continuously hedged by a controlled portfolio of the basic securities. We further study some examples of interest including cumulative barrier options and discuss some numerical implementations.

scholarly journals A large deviation principle for a Brownian immigration particle system

2005 ◽  
Vol 42 (04) ◽  
pp. 1120-1133
Author(s):  
Mei Zhang
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Particle System ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Intensity Measure ◽  
Branching Particle System ◽  
Deviation Principle

We derive a large deviation principle for a Brownian immigration branching particle system, where the immigration is governed by a Poisson random measure with a Lebesgue intensity measure.

scholarly journals Occupation times of alternating renewal processes with Lévy applications

2018 ◽  
Vol 55 (4) ◽  
pp. 1287-1308 ◽  
Author(s):  
Nicos Starreveld ◽  
Réne Bekker ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Levy Processes ◽  
Normal Random Variable ◽  
Occupation Time ◽  
Renewal Processes ◽  
Tail Asymptotics ◽  
Occupation Times

AbstractIn this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

scholarly journals An infinite particle system with additive interactions

1979 ◽  
Vol 11 (02) ◽  
pp. 355-383 ◽  
Author(s):  
Richard Durrett
Keyword(s):  
Necessary Conditions ◽  
Particle System ◽  
Particle Systems ◽  
Compact Set ◽  
Translation Invariant ◽  
Limiting Behavior ◽  
Branching Random Walks ◽  
Sufficient And Necessary Conditions ◽  
Infinite Particle ◽  
Number Of Particles

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set ast→ ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

A LARGE DEVIATION FOR OCCUPATION TIME OF SUPER α-STABLE PROCESS

2008 ◽  
Vol 11 (01) ◽  
pp. 53-71 ◽  
Author(s):  
QIU-YUE LI ◽  
YAN-XIA REN
Keyword(s):  
Brownian Motion ◽  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Stable Process ◽  
Rate Function ◽  
Occupation Time ◽  
Occupation Times ◽  
Tail Probabilities ◽  
Super Brownian Motion

We derive a large deviation principle for occupation time of super α-stable process in ℝd with d > 2α. The decay of tail probabilities is shown to be exponential and the rate function is characterized. Our result can be considered as a counterpart of Lee's work on large deviations for occupation times of super-Brownian motion in ℝd for dimension d > 4 (see Ref. 10).

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