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
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 p...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.展开更多
In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochasti...In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.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
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
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
An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations,whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed t...An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations,whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables.The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained.Numerical simulations are provided.展开更多
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.
文摘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.
基金supported by the National NaturalScience Foundation of China(12071003,11901005)the Natural Science Foundation of Anhui Province(2008085QA20)。
文摘In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.
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.
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the Shanghai Rising-Star Program(Grant No.22QA1406900)the National Natural Science Foundation of China(Grant Nos.11971316,12271368)for their financial supports.
文摘An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations,whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables.The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained.Numerical simulations are provided.
