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 sy...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.展开更多
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...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.展开更多
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 e...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.展开更多
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 ...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.展开更多
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 asympt...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.展开更多
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...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.展开更多
We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Mo...We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Moreover,we prove existence and uniqueness allowing the coefficients in the linear growth-and monotonicity-condition for the generator to be random and time-dependent.In the L2-case with linear growth,this also generalizes the results of Kruse and Popier(Stochastics 88:491–539,2016).For the proof of the comparison result,we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure,we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.展开更多
文摘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.
基金work was supported by National Natural Science Foundation of China(Grant No.11771309)Natural Science and Engineering Research Council of Canada(Grant No.311945-2013)the Fundamental Research Funds for the Central Universities of China。
文摘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.
基金supported by CNRST “Centre National pour la Recherche Scien-tifique et Technique”,No.I003/018,Rabat,Morocco.
文摘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.
文摘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.
基金Guangjun Shen was supported by the Distinguished Young Scholars Foundation of Anhui Province(1608085J06)the Top Talent Project of University Discipline(speciality)(Grant No.gxbjZD03)+2 种基金the National Natural Science Foundation of China(Grant No.11901005)Qian Yu was supported by the ECNU Academic Innovation Promotion Program for Excellent Doctoral Students(YBNLTS2019-010)the Scientific Research Innovation Program for Doctoral Students in Faculty of Economics and Management(2018FEM-BCKYB014).
文摘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.
基金funded by a grant from the Natural Sciences and Engineering Research Council of Canada.
文摘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.
基金Large parts of this article were written when Alexander Steinicke was member of the Institute of Mathematics and Scientific Computing,University of Graz,Austria,and supported by the Austrian Science Fund(FWF):Project F5508-N26,which is part of the Special Research Program"Quasi-Monte Carlo Methods:Theory and Applications."。
文摘We show that the comparison results for a backward SDE with jumps established in Royer(Stoch.Process.Appl 116:1358–1376,2006)and Yin and Mao(J.Math.Anal.Appl 346:345–358,2008)hold under more simplified conditions.Moreover,we prove existence and uniqueness allowing the coefficients in the linear growth-and monotonicity-condition for the generator to be random and time-dependent.In the L2-case with linear growth,this also generalizes the results of Kruse and Popier(Stochastics 88:491–539,2016).For the proof of the comparison result,we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure,we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.