A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was di...A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was divided into two aspects. Firstly, the approximation family was tight using the methods given by Billingsley; secondly, the finite-dimension distributions of approximation family converged weakly to the Rosenblatt process by proving the convergence of the corresponding characteristic functions.展开更多
We study the problem of the approximation in law of the Rosenblatt sheet. We prove the convergence in law of two families of process to the Rosenblatt sheet: the first one is constructed from a Poisson process in the...We study the problem of the approximation in law of the Rosenblatt sheet. We prove the convergence in law of two families of process to the Rosenblatt sheet: the first one is constructed from a Poisson process in the plane and the second one is based on random walks.展开更多
In this work,the optimal control for a class of Hilfer fractional stochastic integrodifferential systems driven by Rosenblatt process and Poisson jumps has been discussed in infinite dimensional space involving the Hi...In this work,the optimal control for a class of Hilfer fractional stochastic integrodifferential systems driven by Rosenblatt process and Poisson jumps has been discussed in infinite dimensional space involving the Hilfer fractional derivative.First,we study the existence and uniqueness of mild solution results are proved by the virtue of fractional calculus,successive approximation method and stochastic analysis techniques.Second,the optimal control of the proposed problem is presented by using Balder’s theorem.Finally,an example is demonstrated to illustrate the obtained theoretical results.展开更多
We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and st...We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.展开更多
In this paper, we are concerned with a class of neutral fractional stochastic partial differential equations driven by a Rosenblatt process. By the stochastic analysis technique, the properties of operator semigroup a...In this paper, we are concerned with a class of neutral fractional stochastic partial differential equations driven by a Rosenblatt process. By the stochastic analysis technique, the properties of operator semigroup and combining the Banach fixed-point theorem, we prove the existence and uniqueness of the mild solutions to this kind of equations driven by Rosenblatt process. In the end, an example is given to demonstrate the theory of our work.展开更多
This manuscript is mainly focusing on the approximate controllability and the null controllability for fractional neutral stochastic partial differential equations with delay driven by Rosenblatt pro-cess.We discuss t...This manuscript is mainly focusing on the approximate controllability and the null controllability for fractional neutral stochastic partial differential equations with delay driven by Rosenblatt pro-cess.We discuss the sufficient conditions for approximate controllability and null controllability for Hilfer fractional neutral stochastic partial differential equations driven by Rosenblatt process.Finally,we provide two examples to verify the obtained results.展开更多
基金Supported by the National Natural Science Foundation of China(11061032)the Fundamental Research Funds for Central Universities HUST(2011QN172)the Hubei Normal University(ZD201118)
基金support of NSF grants(11471105)of China NSF grants(2016CFB526)of Hubei Province Innovation Team of the Educational Department of Hubei Province(T201412)
基金National Natural Science Foundation of China(No. 11171062)Innovation Program of Shanghai Municipal Education Commission,China(No. 12ZZ063)Natural Science Foundation of Bengbu College,China(No. 2010ZR10)
文摘A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was divided into two aspects. Firstly, the approximation family was tight using the methods given by Billingsley; secondly, the finite-dimension distributions of approximation family converged weakly to the Rosenblatt process by proving the convergence of the corresponding characteristic functions.
基金The authors would like to thank the anonymous referees whose remarks and suggestions greatly improved the presentation of the paper. The authors would also like to thank Professor Yimin Xiao, Michigan State University, USA, for stimulating discussion. Guangjun Shen was supported in part by the National Natural Science Foundation of China (Grant No. 11271020) Dongjin Zhu was supported in part by the Key Natural Science Foundation of the Anhui Educational Committee (KJ2012ZD01) and the Philosophy and Social Science Planning Foundation of Anhui Province (AHSKll-12D~28).
文摘We study the problem of the approximation in law of the Rosenblatt sheet. We prove the convergence in law of two families of process to the Rosenblatt sheet: the first one is constructed from a Poisson process in the plane and the second one is based on random walks.
文摘In this work,the optimal control for a class of Hilfer fractional stochastic integrodifferential systems driven by Rosenblatt process and Poisson jumps has been discussed in infinite dimensional space involving the Hilfer fractional derivative.First,we study the existence and uniqueness of mild solution results are proved by the virtue of fractional calculus,successive approximation method and stochastic analysis techniques.Second,the optimal control of the proposed problem is presented by using Balder’s theorem.Finally,an example is demonstrated to illustrate the obtained theoretical results.
文摘We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.
文摘In this paper, we are concerned with a class of neutral fractional stochastic partial differential equations driven by a Rosenblatt process. By the stochastic analysis technique, the properties of operator semigroup and combining the Banach fixed-point theorem, we prove the existence and uniqueness of the mild solutions to this kind of equations driven by Rosenblatt process. In the end, an example is given to demonstrate the theory of our work.
文摘This manuscript is mainly focusing on the approximate controllability and the null controllability for fractional neutral stochastic partial differential equations with delay driven by Rosenblatt pro-cess.We discuss the sufficient conditions for approximate controllability and null controllability for Hilfer fractional neutral stochastic partial differential equations driven by Rosenblatt process.Finally,we provide two examples to verify the obtained results.