Criteria on ergodicity and strong ergodicity of single death processes

Based on an explicit representation of moments of hitting times for single death processes, the criteria on ergodicity and strong ergodicity are obtained. These results can be applied for an extended class of branching processes. Meanwhile, some sufficient and necessary conditions for recurrence and exponential ergodicity as well as extinction probability for the processes are presented.     

Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

We obtain sufficient criteria for central limit theorems (CLTs) for ergodic continuous-time Markov chains (CTMCs). We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.     

Nonlinear branching processes with immigration

The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.     

Unified characteristic numbers and solutions of equations for birth and death processes with barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.     

High order moments of first hitting times for single death processes

We present an explicit and recursive representation for high order moments of the first hitting times of single death processes. Based on that, some necessary or sufficient conditions of exponential ergodicity as well as a criterion on\mathit{\ell }-ergodicity are obtained for single death processes, respectively.     

Moments of first hitting times for birth-death processes on trees

An explicit and recursive representation is presented for moments of the first hitting times of birth-death processes on trees. Based on that, the criteria on ergodicity, strong ergodicity, and l-ergodicity of the processes as well as a necessary condition for exponential ergodicity are obtained.     

Distributions of lifetime after explosion for birth and death processes

How long can a process live after explosion for a birth and death process with explosion and lifespan σ,X ={X(t),t<σ}?Distribution of lifetime from explosion to the end of life, expected lifetime and probability of explosion which means the end of life are found.     

Existence,uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.     

Moments of integral-type downward functionals for single death processes

We get an explicit and recursive representation for high order moments of integral-type downward functionals for single death processes. Meanwhile, main results are applied to more general integral-type downward functionals.     

Some properties of repeated hits after first explosion for birth and death processes

Of repeated hits and repeated explosions after first explosion for a birth and death process with explosion some properties are investigated. The properties of repeated hits after first explosion may be expressed by the properties of the first hit after the first explosion. Project supported by the National Natural Science Foundation of China (Grant No. 19761028) and the Centre of Researching Mathematics and Forstering Higher Talent, the Ministry of Education of China.     

Geometric ergodicity of affine processes on cones

For affine processes on finite-dimensional cones, we give criteria for geometric ergodicity — that is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical term-structure models for credit and interest rate risk.     

Joint distributions of first hitting time and first hitting location after explosion for birth and death processes

For a birth and death processX=|X(t),t <σ| with explosion and lifespanu distributions and joint distributions of first hitting time and first hitting location after explosion of setB _n = |0,1,...,n| ,n have been found.     

On the sample path properties of mixed Poisson processes

The mixed Poisson process has been widely used in financial engineering for modeling arrival of events that cluster in time, as it has strictly stationary and positively correlated increments. However, we show that, surprisingly, the sample autocovariance and autocorrelation of the increments of a mixed Poisson process converge to zero almost surely as the sample size goes to infinity. Consequently, the sample autocovariance or autocorrelation cannot be used in the method of moments for parameter estimation of mixed Poisson processes.     

两类生灭过程的特征数及其概率意义

文章研究以0为反射壁和以0为拟飞射壁的两种生灭过程.通过引入线性方程组和推移算子,证明了朱全新等人(2004)的三个主要结果;准确地定义了mai,eai,Nai,Ra(a≥0)等一系列特征数,并表明了这些特征数的概率意义;最后,求出了Ra<∞(a≥0)的一个等价条件.应该注意的是:杨向群(1986),王梓坤(1996)的一些结果是这里a=0时的特例,因此从某种意义上说,此文是对杨向群(1986),王梓坤(1996)的结果的进一步推广和完善.     

Thinning of point processes,revisited

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广义分枝过程的强遍历性

在文[1]的基础上，进一步讨论了广义分枝Q－矩阵的强遍历性及随机单调性。     

Exit identities for diusion processes observed at Poisson arrival times

For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.     

Classical Method of Moments for Partially and Discretely Observed Ergodic Models

We discuss the method of moments for a partially and discretely observed model driven by a time-homogeneous Lévy process. We suppose that the unobserved process is an -Markov process and that the data, which comes from another process, are available only at regularly spaced time points. Stochastic differential equations are particularly treated among many other possible models. Some illustrative examples are presented with simulations.     

Generation of random processes for fatigue testing

In fatigue testing there is a need to generate multivariate random processes meeting certain conditions on distributions, crossing properties and regularity. In this paper it is demonstrated how these conditions can be satisfied. The required process is generated by a random time deformation working on a Gaussian process.     

一类(QL)型随机微分方程解的轨道唯一性判别及其应用

考虑了一类拟左连续(QL)型随机微分方程(S.D.E.)解的轨道唯一性,应用随机分析方法获得了唯一性成立的一般判别定理,并在方程系数满足局部(或非)Lipschitz条件下给出了一些应用实例.