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.

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.

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.

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.

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

2005 ◽  
Vol 42 (4) ◽  
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 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.

A Criterion of Convergence of Generalized Processes and an Application to a Supercritical Branching Particle System

1991 ◽  
Vol 43 (5) ◽  
pp. 985-997 ◽  
Author(s):  
Begoña Fernández ◽  
Luis G. Gorostiza
Keyword(s):  
Particle System ◽  
Continuous Process ◽  
Integral Functional ◽  
Convergence Criterion ◽  
Convergence In Distribution ◽  
Branching Particle System ◽  
Recent Theorem ◽  
A Minor ◽  
Generalized Processes ◽  
Special Case

The problem of convergence in distribution of a large class of generalized semimartingales to a continuous process is considerably simplified by a recent theorem of Aldous [1], in conjunction with a result of Cremers and Kadelka [3] on convergence of integral functional, and the results of Mitoma [15] and Fouque [8] for generalized processes. We will give a convenient convergence criterion in this setting. The proof amounts to a direct combination of the results of the abovementioned authors, requiring only a minor extension (of a special case) of the theorem of Cremers and Kadelka.

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 BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid Oufdil
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Random Measure ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Stochastic Optimal Control Problem ◽  
Uniqueness Of The Solution ◽  
Logarithmic Growth

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

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