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Zenghu Li

Publications and source records attributed to Zenghu Li.

At least 37 records · Page 2Linked to original sources
arXiv2016

Asymptotic results for exponential functionals of Levy processes

In this work we give a complete description to the asymptotic behaviors of exponential functionals of Lévy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds are proved, the accurate limit constants are also given. As an application, we study the survival probabilities of continuous-state branching processes in random environment defined in He et al. (2016). Like the discrete case and branching diffusion in random environment, we classify them into five different types according to their extinction speeds.

math.PR↗
arXiv2016

Continuous-state branching processes in Levy random environments

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE process with immigration.

math.PR↗
arXiv2014

The maximal jump and local convergence of continuous-state branching processes

We study the distribution of the maximal jump of continuous-state branching processes. Several exact expressions and explicit asymptotics of both the local maximal jump and the global maximal jump are obtained. We also compare the distribution of the maximal jump and the Lévy measure to get several absolute continuity results. Then we study local convergence of continuous-state branching processes under various conditionings. We obtain complete results under the conditioning of large maximal jump, and partial results under two other conditionings, which are, the conditioning of large width, and, the conditioning of large total mass.

math.PR↗
arXiv2014

Yamada-Watanabe results for stochastic differential equations with jumps

Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness imply strong existence for stochastic differential equations with jumps.

math.PR↗
arXiv2014

Fluctuation limits of the super-Brownian motion with a single point catalyst

We prove a fluctuating limit theorem of a sequence of super-Brownian motions over $\mbb{R}$ with a single point catalyst. The weak convergence of the processes on the space of Schwarz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional Brownian motion.

math.PR↗
arXiv2014

Parameter estimation for a subcritical affine two factor model

For an affine two factor model, we study the asymptotic properties of the maximum likelihood and least squares estimators of some appearing parameters in the so-called subcritical (ergodic) case based on continuous time observations. We prove strong consistency and asymptotic normality of the estimators in question.

math.ST↗
arXiv2014

Path-valued branching processes and nonlocal branching superprocesses

A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time-space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637-686] and Abraham and Delmas [Ann. Probab. 40 (2012) 1167-1211].

math.PR↗
arXiv2013

Stationarity and ergodicity for an affine two factor model

We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called alpha-root process with α\in(1,2]. The existence of a unique stationary distribution for the affine process is proved in case of α\in(1,2]; further, in case of α=2, the ergodicity is also shown.

math.PR↗
arXiv2013

On parameter estimation for critical affine processes

First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space $R_+ \times R^d$. We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators of some parameters of a two-dimensional critical affine diffusion process.

math.ST↗
arXiv2013

Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model

We study the estimation of a stable Cox-Ingersoll-Ross model, which is a special subcritical continuous-state branching process with immigration. The process is characterized in terms of some stochastic equations. The exponential ergodicity and strong mixing property of the process and the heavy tail behavior of some related random sequences are studied. We also establish the convergence of some point processes and partial sums associated with the model. From those results, we derive the consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift parameters based on low frequency observations. A weakly consistent estimator is also proposed for the volatility coefficient based on high frequency observations.

math.PR↗
arXiv2012

The reversibility and an SPDE for the generalized Fleming-Viot Processes with mutation

The (Ξ, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.

math.PR↗
arXiv2012

Strong solutions of jump-type stochastic equations

We establish the existence and uniqueness of strong solutions to some jump-type stochastic equations under non-Lipschitz conditions. The results improve those of Fu and Li (2010) and Li and Mytnik (2011).

math.PR↗
arXiv2012

Stochastic equations, flows and measure-valued processes

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming--Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 307--333]. Two scaling limit theorems for the generalized Fleming--Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147--181].

math.PR↗
arXiv2012

Continuous-state branching processes

These notes were used in a short graduate course on branching processes the author gave in Beijing Normal University. The following main topics are covered: scaling limits of Galton--Watson processes, continuous-state branching processes, extinction probabilities, conditional limit theorems, decompositions of sample paths, martingale problems, stochastic equations, Lamperti's transformations, independent and dependent immigration processes. Some of the results are simplified versions of those in the author's book "Measure-valued branching Markov processes" (Springer, 2011). We hope these simplified results will set out the main ideas in an easy way and lead the reader to a quick access of the subject.

math.PR↗
arXiv2012

An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure

This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+β)$-stable branching and Lebesgue initial measure on $\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put τ:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{τ=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/β} dt$ is finite or infinite.

math.PR↗
arXiv2012

Branching Particle Systems in Spectrally One-sided Levy Processes

We investigate the branching structure coded by the excursion above zero of a spectrally positive Levy process. The main idea is to identify the level of the Levy excursion as the time and count the number of jumps upcrossing the level. By regarding the size of a jump as the birth site of a particle, we construct a branching particle system in which the particles undergo nonlocal branchings and deterministic spatial motions to the left on the positive half line. A particle is removed from the system as soon as it reaches the origin. Then a measure-valued Borel right Markov process can be defined as the counting measures of the particle system. Its total mass evolves according to a Crump-Mode-Jagers branching process and its support represents the residual life times of those existing particles. A similar result for spectrally negative Levy process is established by a time reversal approach. Properties of the measure-valued processes can be studied via the excursions for the corresponding Levy processes.

math.PR↗