MOMENTS OF CONTINUOUS-STATE BRANCHING PROCESSES IN LéVY RANDOM ENVIRONMENTS

For continuous-state branching processes in Lévy random environments, the recursion of n-moments and the equivalent condition for the existence of general f-moments are established, where f is a positive continuous function satisfying some standard conditions.     

THE CONSTRUCTION OF MULTITYPE CANONICAL MARKOV BRANCHING CHAINS IN RANDOM ENVIRONMENTS

The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment (CMBCRE) and multitype Markov branching chain in random environment (MBCRE) and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.     

Harmonic moments of branching processes in random environments

We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.     

An L 2-theory for a class of SPDEs driven by Lévy processes

In this paper we present an L 2-theory for a class of stochastic partial differential equations driven by Lévy processes.The coefficients of the equations are random functions depending on time and space variables,and no smoothness assumption of the coefficients is assumed.     

THE EXISTENCE AND MOMENTS OF CANONICAL BRANCHING CHAIN IN RANDOM ENVIRONMENT

The concepts of branching chain in random environmnet and canonical branch-ing chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical expectation and variance of branching chain in random environment are also given.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

Equivalent conditions of local asymptotics for the solutions of defective renewal equations, with applications

The paper obtains some equivalent conditions of local asymptotics for the solutions of defective renewal equations in the heavy-tailed case. As applications, the paper gives a different proof of a classical result about the local distribution of the supremum of a random walk. These results are also applied in examples involving the renewal function for terminating renewal processes and the age-dependent branching processes.     

Moments of Continuous-state Branching Processes with or Without Immigration

For a positive continuous function f satisfying some standard conditions,we study the f-moments of continuous-state branching processes with or without immigration.The main results give criteria for the existence of the f-moments.The characterization of the processes in terms of stochastic equations plays an essential role in the proofs.     

A note on moments of dividends

We reconsider a formula for arbitrary moments of expected discounted dividend payments in a spectrally negative Lévy risk model that was obtained in Renaud and Zhou (2007, [4]) and in Kyprianou and Palmowski (2007, [3]) and extend the result to stationary Markov processes that are skip-free upwards.     

SOME PROPERTIES OF GALTON-WATSON BRANCHING PROCESSES IN VARYING ENVIRONMENTS

This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING （F） OR （B）

This article is a continuation of[9].Based on the discussion of random Kolmogorov forward（backward）equations,for any given q-matrix in random environment, Q（θ）=（q（θ;x,y）,x,y∈X）,an infinite class of q-processes in random environments satisfying the random Kolmogorov forward（backward）equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward（backward）equation are constructed.     

Lindeberg's central limit theorems for martingale-like sequences under sub-linear expectations

The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, the central limit theorem and the functional central limit theorem are obtained for martingale-like random variables under the sub-linear expectation. As applications, the Lindeberg's central limit theorem is obtained for independent but not necessarily identically distributed random variables, and a new proof of the Lévy characterization of a GBrownian motion without using stochastic calculus is given. For proving the results, Rosenthal's inequality and the exponential inequality for the martingale-like random variables are established.     

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT （I）： MODEL,MARKOV PROPERTY,MOMENTS

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment （BCDSTRE）, i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

Heavy tails of a Lévy process and its maximum over a random time interval

Let {Xt,t0} be a Lévy process with Lévy measure ν on(∞,∞),and let τ be a nonnegative random variable independent of {Xt,t0}.We are interested in the tail probabilities of X τ and X(τ) = sup0≤t≤τXt.For various cases,under the assumption that either the Lévy measure ν or the random variable τ has a heavy right tail we prove that both Pr(X τ > x) and Pr(X(τ) > x) are asymptotic to Eτν((x,∞)) + Pr(τ > x/(0 ∨ EX 1)) as x →∞,where Pr(τ > x/0) = 0 by convention.     

ELEMENTARY BIFURCATIONS FOR A SIMPLE DYNAMICAL SYSTEM UNDER NON-GAUSSIAN LEVY NOISES

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies. A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian α-stable Lévy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by solving a nonlocal Fokker-Planck equation numerically. This allows numerically investigating phenomenological bifurcation, or P-bifurcation, for stochastic differential equations with non-Gaussian Lévy noises.     

Option Pricing for Time-Change Exponential Lévy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price whenthe underlying price process is described by a time-change Lévy process.European option pricing formula isobtained under the minimal entropy martingale measure(MEMM)and applied to several examples of particulartime-change Lévy processes.It can be seen that the framework in this paper encompasses the Black-Scholesmodel and almost all of the models proposed in the subordinated market.     

Smooth-pasting property on reflected Lévy processes and its applications in credit risk modeling

We study the smooth-pasting property for a class of conditional expectations with reflected Lévy process as underlying state process.A relationship between local times and regulators for the doubly reflected Lévy process is established.As applications,we derive the analytic pricing formula for a zero-coupon defaultable bond when the default intensity(resp.the stochastic loss rate)is modeled as one-sided(resp.double-sided)reflected Lévy processes.Finally,some numerical illustrations are provided.     

RENEWAL THEOREM FOR （L, 1）-RANDOM WALK IN RANDOM ENVIRONMENT

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.     

Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper,we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes.For a test,we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations.It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge.The test statistics are evaluated by some empirical results.     

Exponential convergence in probability for empirical means of Lévy processes

Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.