In this paper,we prove some limsup results for increments and lag increments of G(t),which is a stable processe in random scenery.The proofs rely on the tail probability estimation of G(t).
In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient cond...In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder continuity.Then we show that this condition is also necessary for the existence of derivatives of intersection local time at the origin.Moreover,we also study the power variation of the derivatives.展开更多
In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued...In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.展开更多
Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical C...Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X（t） are established.展开更多
Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic diff...Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic differential equation dXt = σ（Xt-）dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.展开更多
In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend t...In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.展开更多
The alpha stable self-similar stochastic process has been proved an effective model for high variable data traffic. A deep insight into some special issues and considerations on use of the process to model aggregated ...The alpha stable self-similar stochastic process has been proved an effective model for high variable data traffic. A deep insight into some special issues and considerations on use of the process to model aggregated VBR video traffic is made. Different methods to estimate stability parameter a and self-similar parameter H are compared. Processes to generate the linear fractional stable noise （LFSN） and the alpha stable random variables are provided. Model construction and the quantitative comparisons with fractional Brown motion （FBM） and real traffic are also examined. Open problems and future directions are also given with thoughtful discussions.展开更多
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 a...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.展开更多
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 se...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.展开更多
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 proc...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.展开更多
Let Y_(t) be an autoregressive process with order one,i.e.,Y_(t)=μ+ϕnY_(t-1)+εt,where fεtg is a heavy tailed general GARCH noise with tail indexα.Letϕn be the least squares estimator(LSE)ofϕn.Forμ=0 andα<2,it...Let Y_(t) be an autoregressive process with order one,i.e.,Y_(t)=μ+ϕnY_(t-1)+εt,where fεtg is a heavy tailed general GARCH noise with tail indexα.Letϕn be the least squares estimator(LSE)ofϕn.Forμ=0 andα<2,it is shown by Zhang and Ling(2015)thatϕn is inconsistent when Y_(t) is stationary(i.e.,ϕn.,ϕ<1),however,Chan and Zhang(2010)showed thatϕn is still consistent with convergence rate n when Y_(t) is a unit-root process(i.e.,ϕn=1)and fεtg is a GARCH(1,1)noise.There is a gap between the stationary and nonstationary cases.In this paper,two important issues will be considered:(1)what about the nearly unit root case?(2)When canϕbe estimated consistently by the LSE?We show that whenϕn=1-c/n,then bϕn converges to a functional of stable process with convergence rate n.Further,we show that if limn!1 kn(1-ϕn)=c for a positive constant c,then kn(ϕn-ϕ)converges to a functional of two stable variables with tail indexα/2,which means thatϕn can be estimated consistently only when kn!1.展开更多
Based on macroscopic and synthetic approaches, especially information entropy approach, the quantification of the flexible degree and order degree of business processes is studied. According to the outcome of above an...Based on macroscopic and synthetic approaches, especially information entropy approach, the quantification of the flexible degree and order degree of business processes is studied. According to the outcome of above analysis, a conceptual model of optimizing business processes is proposed which supports to construct dynamic stable business processes. The research above has been applied in project 863/SDDAC-CIMS, and achieved primary benefits.展开更多
We consider the exponentially ergodic properties of systems of SDEs in Rndriven by cylindrical stable processes,potentially with different indices across different coordinates.Our approach is based on the well-known F...We consider the exponentially ergodic properties of systems of SDEs in Rndriven by cylindrical stable processes,potentially with different indices across different coordinates.Our approach is based on the well-known Foster-Lyapunov criteria and a careful selection of Lyapunov functions,alongside recent advances in regularity and transition density estimates for solutions to SDEs driven by Lévy processes with independent coordinates.These results are novel,even in the one-dimensional case.Notably,our findings suggest that multiplicative cylindrical stable processes can enhance the ergodicity of the system when the stable noise indices in all directions fall within[1,2).展开更多
The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satis...The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies thespace-fractional telegraph equation, is presented. Its limiting behaviour and the connectionwith symmetric stable processes is also examined.展开更多
Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy proc...Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy processes(1/2<α≤1),where the drift coefficient is Holder continuous in space variable,while the noise coeficient is Lipscitz continuous in space variable,and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance.If the drift coefficient does not depend on distribution variable,our methodology developed in this paper applies to the caseαe(0,1].The main tool relies on heat kernel estimates for(distribution independent)stable SDEs and Banach's fixed point theorem.展开更多
Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions f...Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.展开更多
In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is hon...In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.展开更多
We study the functional limits of continuous-time random walks （CTRWs） with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an a-stable Levy process in D（[0, 1]） with M1...We study the functional limits of continuous-time random walks （CTRWs） with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an a-stable Levy process in D（[0, 1]） with M1-topology but the corresponding scaled CTRWs converge weakly to the same limit in D（[0, 1]） with J1-topology.展开更多
文摘In this paper,we prove some limsup results for increments and lag increments of G(t),which is a stable processe in random scenery.The proofs rely on the tail probability estimation of G(t).
基金Supported by National Natural Science Foundation of China(Grant Nos.12071003,12201294)Natural Science Foundation of Jiangsu Province,China(Grant No.BK20220865)。
文摘In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder continuity.Then we show that this condition is also necessary for the existence of derivatives of intersection local time at the origin.Moreover,we also study the power variation of the derivatives.
基金Supported partly by the NNSF of China(Nos.10371092,10171015 and No.10271027)
文摘In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10131040, 10371109 and 10801118)
文摘Let {X（t）,t ∈ R＋} be an integrated α stable process. In this paper, a functional law of the iterated logarithm （LIL） is derived via estimating the small ball probability of X. As a corollary,, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X（t） are established.
基金The authors were indebted to the referees for their helpful comments and careful corrections. The first author＇s work was supported by the Key Laboratory of Random Complex Structures and Data Sciences, Chinese Academy of Sciences （2008DP173182）, the National Natural Science Foundation of China （Grant No. 11571347）, and Academy of Mathematics and Systems Science （Y129161ZZ1）. The second author＇s work was supported by the National Natural Science Foundation of China （Grant Nos. 11201073 and 11522106）, the National Science Foundation of Fujian Province （2015J01003）, and the Program for Nonlinear Analysis and Its Applications （IRTL1206）.
文摘Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic differential equation dXt = σ（Xt-）dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.
基金supported by National Natural Science Foundation of China (10901054)
文摘In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.
文摘The alpha stable self-similar stochastic process has been proved an effective model for high variable data traffic. A deep insight into some special issues and considerations on use of the process to model aggregated VBR video traffic is made. Different methods to estimate stability parameter a and self-similar parameter H are compared. Processes to generate the linear fractional stable noise （LFSN） and the alpha stable random variables are provided. Model construction and the quantitative comparisons with fractional Brown motion （FBM） and real traffic are also examined. Open problems and future directions are also given with thoughtful discussions.
基金Supported by the National Natural Science Foundation and the Doctoral Programme Foundation of China.
文摘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.
基金Supported by the National Natural Science Foundation of China
文摘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.
基金the National Natural Science Foundation of China
文摘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.
基金supported by the National Natural Science Foundation of China(11771390, 12171427)ZPNSFC(LZ21A010002)+2 种基金Fundamental Research Funds for the Central Universities (2021XZZX002)supported by Natural Science Foundation of Fujian Province(2020J01794)Fujian Key Laboratory of Granular Computing and Applications(Minnan Normal University)
文摘Let Y_(t) be an autoregressive process with order one,i.e.,Y_(t)=μ+ϕnY_(t-1)+εt,where fεtg is a heavy tailed general GARCH noise with tail indexα.Letϕn be the least squares estimator(LSE)ofϕn.Forμ=0 andα<2,it is shown by Zhang and Ling(2015)thatϕn is inconsistent when Y_(t) is stationary(i.e.,ϕn.,ϕ<1),however,Chan and Zhang(2010)showed thatϕn is still consistent with convergence rate n when Y_(t) is a unit-root process(i.e.,ϕn=1)and fεtg is a GARCH(1,1)noise.There is a gap between the stationary and nonstationary cases.In this paper,two important issues will be considered:(1)what about the nearly unit root case?(2)When canϕbe estimated consistently by the LSE?We show that whenϕn=1-c/n,then bϕn converges to a functional of stable process with convergence rate n.Further,we show that if limn!1 kn(1-ϕn)=c for a positive constant c,then kn(ϕn-ϕ)converges to a functional of two stable variables with tail indexα/2,which means thatϕn can be estimated consistently only when kn!1.
文摘Based on macroscopic and synthetic approaches, especially information entropy approach, the quantification of the flexible degree and order degree of business processes is studied. According to the outcome of above analysis, a conceptual model of optimizing business processes is proposed which supports to construct dynamic stable business processes. The research above has been applied in project 863/SDDAC-CIMS, and achieved primary benefits.
基金supported by the National Key R&D Program of China(Grant No.2022YFA1006003)National Natural Science Foundation of China(Grant Nos.11831014,12071076,and 12225104)。
文摘We consider the exponentially ergodic properties of systems of SDEs in Rndriven by cylindrical stable processes,potentially with different indices across different coordinates.Our approach is based on the well-known Foster-Lyapunov criteria and a careful selection of Lyapunov functions,alongside recent advances in regularity and transition density estimates for solutions to SDEs driven by Lévy processes with independent coordinates.These results are novel,even in the one-dimensional case.Notably,our findings suggest that multiplicative cylindrical stable processes can enhance the ergodicity of the system when the stable noise indices in all directions fall within[1,2).
基金Project supported by the National Natural Science Foundation of China (No. 10071014).
文摘The space-fractional telegraph equation is analyzed and the Fourier transform of its funda-mental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies thespace-fractional telegraph equation, is presented. Its limiting behaviour and the connectionwith symmetric stable processes is also examined.
文摘Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy processes(1/2<α≤1),where the drift coefficient is Holder continuous in space variable,while the noise coeficient is Lipscitz continuous in space variable,and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance.If the drift coefficient does not depend on distribution variable,our methodology developed in this paper applies to the caseαe(0,1].The main tool relies on heat kernel estimates for(distribution independent)stable SDEs and Banach's fixed point theorem.
基金supported by the National Natural Science Foundation(Grant No.10671188)Special Foundation of USTC
文摘Let {X（t）, t ≥ 0} be a Lévy process with EX（1） = 0 and EX^2（1） 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X（t）, t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.
基金Xiangtan University New Staff Research Start-up Grant (Grant No. 08QDZ27)
文摘In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.
基金Acknowledgements The author thanks Professor Yimin Xiao for stimulating discussion. Thanks are also due to the anonymous referees for their careful reading and useful suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 10901054）.
文摘We study the functional limits of continuous-time random walks （CTRWs） with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an a-stable Levy process in D（[0, 1]） with M1-topology but the corresponding scaled CTRWs converge weakly to the same limit in D（[0, 1]） with J1-topology.
