Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Levy Processes

This paper first develops a Lyapunov-type theorem to study global well-posedness(existence and uniqueness of the strong variational solution)and asymptotic stability in probability of nonlinear stochastic evolution systems(SESs)driven by a special class of Levy processes,which consist of Wiener and compensated Poisson processes.This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes.The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system,and minimizes a meaningful cost functional that penalizes both states and control.The approach does not require to solve a Hamilton-Jacobi-Bellman equation(HJBE).An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

Polarity for a Class of Levy Processes

Polar set of Markov processes is an important concept in probabilistic potential theory, but it is not easy to judge the polarity of the sets. In this paper, we give some results which can be easily used to examine the polarity of the sets whenX t belongs to a special class of Levy processes. We also give a result about polar functions of symmetric stable processes.

Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Levy processes with periodic mean

VVc deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Levy process.For this estimator,we obtain consistency and the asymptotic distribution.Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Levy process,they can be regarded both as a Levy generalization of fractional Brownian motion and a fractional generalization of Levy process.

Ito Formula for Integral Processes Related to Space-Time Levy Noise

In this article, we give a new proof of the Itôformula for some integral processes related to the space-time Lévy noise introduced in [1] [2] as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the p-th moment of the stochastic integral, and the Itôrepresentation theorem leading to a chaos expansion similar to the Gaussian case.

STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

POLYNOMIAL RECURRENCE FOR LEVY PROCESSES

In this paper, the authors study the ω-transience and ω-recurrence for Levy processes with any weight function ω, give a relation between ω-recurrence and the last exit times. As a special case, the polynomial recurrence and polynomial transience are also studied.

α-TRANSIENCE AND α-RECURRENCE FOR RANDOM WALKS AND LEVY PROCESSES

The authors investigate the α-transience and α-recurrence for random walks and Levy processes by means of the associated moment generating function, give a dichotomy theorem for not one-sided processes and prove that the process X is quasisymmetric if and only if X is not α-recurrent for all α＜ 0 which gives a probabilistic explanation of quasi-symmetry, a concept originated from C. J. Stone.

A stochastic threshold of a delayed epidemic model incorporating Levy processes with harmonic mean and vaccination

The aim of this paper is to investigate a stochastic threshold for a delayed epidemic model driven by Levy noise with a nonlinear incidence and vaccination.Mainly,we derive a stochastic threshold 77s which depends on model parameters and stochastic coefficients for a better understanding of the dynamical spreading of the disease.First,we prove the well posedness of the model.Then,we study the extinction and the persistence of the disease according to the values of TZS.Furthermore,using different scenarios of Tuberculosis disease in Morocco,we perform some numerical simulations to support the analytical results.

Hunt’s Hypothesis(H)for the Sum of Two Independent Levy Processes

Which Levy processes satisfy Hunt’s hypothesis(H)is a long-standing open problem in probabilistic potential theory.The study of this problem for one-dimensional Levy processes suggests us to consider(H)from the point of view of the sum of Levy processes.In this paper,we present theorems and examples on the validity of(H)for the sum of two independent Levy processes.We also give a novel condition on the Levy measure which implies(H)for a large class of one-dimensional Levy processes.

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 （CMJ） 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.

A stochastic vaccinated epidemic model incorporating Levy processes with a general awareness-induced incidence

In this paper,we investigate a stochastic vaccinated epidemic model with a general awareness-induced incidence perturbed by Levy noise.First,we show that this model has a unique global positive solution.Therefore,we establish the dynamic behavior of the solution around both disease-free and endemic equilibrium points.Furthermore,when R_(0)>1,we give sufficient conditions for the existence of an ergodic stationary distribution to the model when the jump part in the Levy noise is null.Finally,we present some examples to illustrate the analytical results by numerical simulations.

LOCAL TIME ANALYSIS OF ADDITIVE LVY PROCESSES WITH DIFFERENT LVY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.

HOMEOMORPHISM FLOWS FOR NON-LIPSCHITZ SDES DRIVEN BY LVY PROCESSES

In this article, homeomorphism flows for non-Lipschitz stochastic differential equations driven by Levy processes are studied.

SELF-INTERSECTION LOCAL TIME OF ADDITIVE LEVY PROCESS

This article discusses the problem of existence of jointly continuous self-intersection local time for an additive levy process. Here, 'local time' is understood in the sense of occupation density, and by an additive Levy process the authors mean a process X = {X(t),t∈ R+N} which has the decomposition X = Xi X2 … XN, each Xl has the lower index αl, α= min{α1,…, αN}. Let Z = (Xt2 - Xt1, …, Xtr - Xtr-1). They prove that if Nrα > d(r-1), then a jointly continuous local time of Z, i.e. the self-intersection local time of X, can be obtained.

MOMENTS OF PASSAGE TIMES AND ASYMPTOTIC BEHAVIOR OF INCREASING SELF-SIMILAR MARKOV PROCESSES

By using Lamperti＇s bijection between self-similar Markov processes and L@vy processes~ we prove finiteness of moments and asymptotic behavior of passage times for increasing self-similar Markov processes valued in （0, ~）. We Mso investigate the behavior of the process when it crosses a level. A limit theorem concerning the distribution of the process immediately before it crosses some level is proved. Some useful examples are given.

Local Time of Additive Levy Process

We studied the problem of existence of jointly continuous local time for an additive process. Here, 'local time' is understood in the sence of occupation density, and by an additive Levy process we mean a process X = {X(t), t ∈ R^d_+ ) } which has the decomposition X= X_1, X_2 ... X_N. We prove that if the product of it slower index and N is greater than d, then a jointly continuous local time can he obtained via Berman's method.

First exit distribution and path continuity of Hunt processes

This paper gives a characterization of a Hunt process path by the first exit left limit distribution. It is also showed that if the first exit left limit distribution leaving any ball from the center is a uniform distribution on the sphere, then the Levy Processes are a scaled Brownian motion.

DIMENSION RESULTS FORORNSTEIN-UHLENBECK TYPE MARKOVPROCESS

Let {X-t, t greater than or equal to 0} be an Ornstein-Uhlenbeck type Markov process with Levy process A(t), the authors consider the fractal properties of its ranges, give the upper and lower bounds of the Hausdorff dimensions of the ranges and the estimate of the dimensions of the level sets for the process. The existence of local times and occupation times of X-t are considered in some special situations.

The Multifractal Analysis of the Occupation Measure of a Lévy Process

We introduce the results on the multifractal structure of the occupation measures of a Brownian Motion, a stable process, a general subordinator and a stochastic process derived from random reordering of the Cantor set. We also introduced an interesting and powerful technique to investigate the multifractal spectrum.

Successful couplings for a class of stochastic differential equations driven by Lvy processes

By constructing proper coupling operators for the integro-differential type Markov generator, we establish the existence of a successful coupling for a class of stochastic differential equations driven by Levy processes. Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups, and it is sharp for Ornstein-Uhlenbeck processes driven by s-stable Levy processes.