人口数相依的两性分支过程的极限性质

本文研究了后代分布依赖于人口数的两性Galton-Watson分支过程, 在对后代分布的适当假设下,对于上临界的情况, 我们研究了有关过程的几乎处处收敛的极限性质.     

Population Size Dependent Generalized Multitype Branching Processes

Abstract

In this paper, we introduce the population size dependent generalized multitype branching process. This is a Markovian model that allows us to study homogeneous multitype branching processes in a unified way. The basic properties for this model, transitions between its states, as well as the existence of a stationary limiting distribution, are investigated. Finally, we apply the obtained results to a new controlled multitype branching process.     

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

Asymptotic Behavior of Critical Primitive Multi-Type Branching Processes with Immigration

Under natural assumptions a Feller-type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly toward a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.     

The ‘birth-and-assassination’ process

A new variant of a branching process is introduced, with sufficient conditions for it to persist and to die out. The model is applied to discuss the asymptotic stability of a new type of queuing process.     

On long time almost sure asymptotics of renormalized branching diffusion processes

We are concerned with the long time behavior of branching diffusion processes. We give a partial answer to the following question: given a smooth density g₀, a branching rate and a spatial motion, does there exist a (nonspatially homogeneous) binary offspring distribution such that the corresponding renormalized branching process tends a.s. to g₀(y) dy as time grows to infinity?     

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.     

The existence of moments of the conditioned limit of the subcritical Crump-Mode-Jagers branching process

If Z (t) is the sum of the characteristics at time t of the population in a Crump-Mode-Jagers branching process, and T is the time to extinction, it is known that under certain conditions, the distribution of Z (t) conditioned on {T > t} converges to a proper distribution as t→∞. We derive necessary and sufficient conditions in terms of the offspring process, for the existence of integral moments of this limit distribution.     

Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps

We consider Markov processes built from pasting together pieces of strong Markov processes which are killed at a position dependent rate and connected via a transition kernel. We give necessary and sufficient conditions for local absolute continuity of probability laws for such processes on a suitable path space and derive an explicit formula for the corresponding likelihood ratio process. The main tool is the consideration of the process between successive jumps – what we call ‘elementary experiments’ – and criteria for absolute continuity of laws of the process there. We apply our results to systems of branching diffusions with interactions and immigrations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Branching/queueing networks: Their introduction and near-decomposability asymptotics

A new class of models, which combines closed queueing networks with branching processes, is introduced. The motivation comes from MIMD computers and other service systems in which the arrival of new work is always triggered by the completion of former work, and the amount of arriving work is variable. In the variant of branching/queueing networks studied here, a customer branches into a random and state-independent number of offspring upon completing its service. The process regenerates whenever the population becomes extinct. Implications for less rudimentary variants are discussed. The ergodicity of the network and several other aspects are related to the expected total number of progeny of an associated multitype Galton-Watson process. We give a formula for that expected number of progeny. The objects of main interest are the stationary state distribution and the throughputs. Closed-form solutions are available for the multi-server single-node model, and for homogeneous networks of infinite-servers. Generally, branching/queueing networks do not seem to have a product-form state distribution. We propose a conditional product-form approximation, and show that it is approached as a limit by branching/queueing networks with a slowly varying population size. The proof demonstrates an application of the nearly complete decomposability paradigm to an infinite state space. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Critical Branching Process with Stationary-Limiting Distribution

Abstract

Seneta (Seneta, E. The stationary distribution of a branching process allowing immigration: A remark on the critical case. J. Royal Statistical Society, Series B 1968, 30, 176–179) shows that a critical branching process with pure immigration has a stationary-limiting distribution provided that its offspring variance is infinite. We obtain a stationary-limiting distribution keeping the variance finite but allowing an emigration–immigration component in each generation.     

出生率与年龄段有关的生灭分枝树

本文基于生灭过程的生灭演化机理, 将生物繁衍过程描述为有向随机图过程-随机分枝树, 建立了出生率与年龄段有关的生灭分枝树演化模型. 本文研究了任一节点在不同年龄及临死时刻的出度分布、虚出度分布和拟出度分布, 并证明了拟出度过程是随机时刻终止的Poisson过程, 讨论了首生年龄及相对出生年龄的分布, 给出了任一节点成为孤立节点的概率.     

A new approach in analyzing extinction probability of Markov branching process with immigration and migration

We use a new approach to consider the extinction properties of the Markov branching process with immigration and migration recently discussed by Li and Liu [Sci. China Math., 2011, 54: 1043–1062]. Some much better explicit expressions are obtained for the extinction probabilities of the subtle super-interacting case.     

Infinite-Horizon Stochastic Differential Games with Branching Payoffs

In this paper, we consider infinite-horizon stochastic differential games with an autonomous structure and steady branching payoffs. While the introduction of additional stochastic elements via branching payoffs offers a fruitful alternative to modeling game situations under uncertainty, the solution to such a problem is not known. A theorem on the characterization of a Nash equilibrium solution for this kind of games is presented. An application in renewable resource extraction is provided to illustrate the solution mechanism.     

Optimal Control and Filtering of the Reproduction Law of a Branching Process

A finite horizon control problem for the reproduction law of a branching process is studied. Some examples with complete information are tackled via the Hamilton–Jacobi–Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and the observations and the observation process has unbounded intensity, a Girsanov-type change of probability measure can be set and the filtering equation for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, also for the Kushner–Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitting process and the unnormalized conditional distribution. By the strong uniqueness for the Zakai equation, equivalence between the partially observable control problem and the separated one is proved.     

Identities for stopped markov chains and their applications

In this paper we obtain identities for some stopped Markov chains. These identities give a unified approach to many problems in optimal stopping of a Markovian sequence, extinction probability of a Markovian branching process and martingale theory.     

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首先, 当$Q$是一个拟单调的q矩阵的时候, 我们找出最小的$Q$函数是一个Feller的转移函数的准则. 然后我们把这个结论应用于生成分支q矩阵并得到相应的生成分支过程的Feller准则. 特别地, 设$\theta$是分支q矩阵中的非线性数, 总是存在一个分点$\theta_0$满足$1\leq\theta_0\leq2$或$\theta_0<+\infty$使得 生成分支过程是否是Feller的要依据$\theta<\theta_0$或者$\theta>\theta_0$.     

Bounds of deviation for branching chains in random environments

We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the process.