In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn＋1（[0, T ]n） is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of...In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn＋1（[0, T ]n） is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of interpolation approximation of fractional Brownian motion BtH with Hurst parameter H 1/2. The limit process is the multiple Stratonovich integral of the function f . In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is 2H in the mean square sense, where is the norm of the partition generating the approximations.展开更多
A Monte Carlo implicit simulation program,Implicit Stratonovich Stochastic Differential Equations(ISSDE),is developed for solving stochastic differential equations(SDEs)that describe plasmas with Coulomb collision.The...A Monte Carlo implicit simulation program,Implicit Stratonovich Stochastic Differential Equations(ISSDE),is developed for solving stochastic differential equations(SDEs)that describe plasmas with Coulomb collision.The basic idea of the program is the stochastic equivalence between the Fokker-Planck equation and the Stratonovich SDEs.The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision.The cases of Lorentzian plasma,Maxwellian plasma and arbitrary distribution function of background plasma have been considered.The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge-Kutta methods.ISSDE is built with C++and has standard interfaces and extensible modules.The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE,which shows its correctness and reliability.展开更多
In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-Itöformula, we obtain the relationship between Hermite polynomials a...In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-Itöformula, we obtain the relationship between Hermite polynomials and multiple G-Stratonovich integrals by using mathematical induction method.展开更多
The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term,where the input is a random input and the state is the output of dynamical systems.Stochastically influenced bilinear sy...The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term,where the input is a random input and the state is the output of dynamical systems.Stochastically influenced bilinear systems are described via bilinear stochastic differential equations.In this paper,first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itôcounterpart.Second,the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit.The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well.The theory of this paper hinges on the‘Stratonovich calculus’,conversion of the Stratonovich integral into the Itôintegral and characteristic function of the vector Brownian motion.The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.展开更多
In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion i...In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.展开更多
基金supported by the scientific research fund of Central South University for Nationalities (YZZ09005)
文摘In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn＋1（[0, T ]n） is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of interpolation approximation of fractional Brownian motion BtH with Hurst parameter H 1/2. The limit process is the multiple Stratonovich integral of the function f . In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is 2H in the mean square sense, where is the norm of the partition generating the approximations.
基金Project supported by the National MCF Energy R&D Program of China(Grant No.2018YFE0304100)the National Key Research and Development Program of China(Grant Nos.2016YFA0400600,2016YFA0400601,and 2016YFA0400602)the National Natural Science Foundation of China(Grant Nos.NSFC-11805273 and NSFC-11905220).
文摘A Monte Carlo implicit simulation program,Implicit Stratonovich Stochastic Differential Equations(ISSDE),is developed for solving stochastic differential equations(SDEs)that describe plasmas with Coulomb collision.The basic idea of the program is the stochastic equivalence between the Fokker-Planck equation and the Stratonovich SDEs.The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision.The cases of Lorentzian plasma,Maxwellian plasma and arbitrary distribution function of background plasma have been considered.The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge-Kutta methods.ISSDE is built with C++and has standard interfaces and extensible modules.The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE,which shows its correctness and reliability.
文摘In this paper, we propose the multiple Stratonovich integral driven by G-Brownian motion under the G-expectation framework. Then based on G-Itöformula, we obtain the relationship between Hermite polynomials and multiple G-Stratonovich integrals by using mathematical induction method.
文摘The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term,where the input is a random input and the state is the output of dynamical systems.Stochastically influenced bilinear systems are described via bilinear stochastic differential equations.In this paper,first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itôcounterpart.Second,the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit.The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well.The theory of this paper hinges on the‘Stratonovich calculus’,conversion of the Stratonovich integral into the Itôintegral and characteristic function of the vector Brownian motion.The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.
基金partially supported by NNSF of China (60534080)the firstauthor is supported in part by the National Science Foundation (DMS0504783)
文摘In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.
