In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H...In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.展开更多
Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)...Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.展开更多
In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the ...In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.展开更多
Let B^(a,b)be a weighted-fractional Brownian motion with Hurst indices a and b such that a>-1 and 0≤b<1∧(1+a).In this paper,we consider the linear self-attracting diffusion dX_(t)^(a,b)=dB_(t)^(a,b)−θ(∫t 0(X...Let B^(a,b)be a weighted-fractional Brownian motion with Hurst indices a and b such that a>-1 and 0≤b<1∧(1+a).In this paper,we consider the linear self-attracting diffusion dX_(t)^(a,b)=dB_(t)^(a,b)−θ(∫t 0(X_(t)^(a,b)−X_(s)^(a,b))ds)dt+νdt with X_(0)^(a,b),whereθ>0 andν∈R are two real parameters.The model is an analog of the linear selfinteracting diffusion(see Cranston and Le Jan(1995)).Under the continuous observation,we study asymptotic behaviors of the least squares estimatorsθˆT andνˆT.In particular,when b>1/2,we obtain a new random variable Z_(1)^(a,b)which is called the Rosenblatt random variable if a=0,and we show that C_(a,b)T^(2-2b)(θ_(T)-θ)converges in distribution to the sum of the chi-square random variable with 1 degree of freedom and the random variable Z_(1)^(a,b).展开更多
Let B^(H)={B_(t)^(H),t≥0}be a fractional Brownian motion with Hurst index 0<H<1,and let B={B_(t),t≥0}be an independent Brownian motion.In this study,we investigate the parameter estimation of a mixed fractiona...Let B^(H)={B_(t)^(H),t≥0}be a fractional Brownian motion with Hurst index 0<H<1,and let B={B_(t),t≥0}be an independent Brownian motion.In this study,we investigate the parameter estimation of a mixed fractional Black-Scholes modelS_(t)^(H)=S_(0)^(H)+μ∫_(0)^(t)S_(s)^(H)ds+σ∫_(0)^(t)S_(s)^(H)d(B_(s)+B_(s)^(H)),whereσ>0,μ∈Rare two unknown parameters.Using quasi-likelihood estimation,when the system is observed at some discrete time instants{t_(i)=ih,i=0,1,2,…,n},we give estimations of the parametersμandσprovided h=h(n)→0 nh→∞and h^(1+γ)n→1for someγ>0,as n→∞.We present the asymptotic normality of the estimators based on the velocity of nh^(1+γ)-1 tending to zero as n tends to infinity.Finally,we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.展开更多
Let BH={BHt,t≥0}be a fractional Brownian motion with Hurst index H∈(0,1).Inspired by pathwise integrals and Wick product,in this paper,we consider the forward and symmetric Wick-Itôintegrals with respect to BH ...Let BH={BHt,t≥0}be a fractional Brownian motion with Hurst index H∈(0,1).Inspired by pathwise integrals and Wick product,in this paper,we consider the forward and symmetric Wick-Itôintegrals with respect to BH as follows:∫t0us⋄d−BHs=limε↓01ε∫t0us⋄(BHs+ε−BHs)ds,∫t0us⋄d∘BHs=limε↓012ε∫t0us⋄(BHs+ε−BH(s−ε)∨0)ds,in probability,where◊denotes the Wick product.We show that the two integrals coincide with divergence-type integral of BH for all H∈(0,1).展开更多
Let B^H1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence Sn：=∑i=0^n-1K...Let B^H1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence Sn：=∑i=0^n-1K（n^αBi^H,K1）（Bi＋1^H2,K2-Bi^H2,K2）where K is a standard Gaussian kernel function and the bandwidth parameter α satisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion B^H1,K1. We also give the stable convergence of the sequence Sn by using the techniques of the Malliavin calculus.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
Let B^(H)be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the self-repelling diffusion X_(t)^(H)=B_(t)^(H)+∫_(0)^(t)∫_(0)^(s)g(u)dX_(u)^(H)ds+vt.wherc v∈R,and g is a nonncgativ...Let B^(H)be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the self-repelling diffusion X_(t)^(H)=B_(t)^(H)+∫_(0)^(t)∫_(0)^(s)g(u)dX_(u)^(H)ds+vt.wherc v∈R,and g is a nonncgative Borel function.The process is an analogue of lincar self-interacting diffusion(M.Cranston and Y.Le Jan Math.Ann.303(1995).87-93.).Based on the asymptotic behavior of the weight function g at infinity,we establish the large time behavior of the recursive convergence of the solution xH.For example,when g∈C^(∞)(R_(+))and 0<g(t)→+∞(t→+∞)。we demonstrate that there is a sequcuce{λ_(n)}of positive real numbers such that J_(t)^(H)(0:g):=g(t)c^(-G(t))X_(t)^(H)→ν+ξ_(∞)^(H)and J_(t)^(H)(0:g):=G(t)(J_(t)^(H)(n-1:g)-λ_(n-1)(cs_(∞)^(H)+ν))→λ_(n)(cs_(∞)^(H)+ν)(t→+∞)in L^(2)and almost surely for every n∈{1,2....}.where G(t)=∫_(0)^(t)g(s)ds and cS_(∞)^(H):=∫_(0)^(∞)g(r)c^(-G(r))dB_(r)^(H).展开更多
基金The research of L.Yan was partially supported bythe National Natural Science Foundation of China (11971101)The research of Z.Chen was supported by National Natural Science Foundation of China (11971432)+3 种基金the Natural Science Foundation of Zhejiang Province (LY21G010003)supported by the Collaborative Innovation Center of Statistical Data Engineering Technology & Applicationthe Characteristic & Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics)the First Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics)。
文摘In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.
文摘Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.
文摘In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.
基金supported by National Natural Science Foundation of China(Grant No.11971101)。
文摘Let B^(a,b)be a weighted-fractional Brownian motion with Hurst indices a and b such that a>-1 and 0≤b<1∧(1+a).In this paper,we consider the linear self-attracting diffusion dX_(t)^(a,b)=dB_(t)^(a,b)−θ(∫t 0(X_(t)^(a,b)−X_(s)^(a,b))ds)dt+νdt with X_(0)^(a,b),whereθ>0 andν∈R are two real parameters.The model is an analog of the linear selfinteracting diffusion(see Cranston and Le Jan(1995)).Under the continuous observation,we study asymptotic behaviors of the least squares estimatorsθˆT andνˆT.In particular,when b>1/2,we obtain a new random variable Z_(1)^(a,b)which is called the Rosenblatt random variable if a=0,and we show that C_(a,b)T^(2-2b)(θ_(T)-θ)converges in distribution to the sum of the chi-square random variable with 1 degree of freedom and the random variable Z_(1)^(a,b).
基金supported by the National Natural Science Foundation of China(Grant Nos.11971101 and 12171081)Shanghai Natural Science Foundation(Grant No.24ZR1402900).
文摘Let B^(H)={B_(t)^(H),t≥0}be a fractional Brownian motion with Hurst index 0<H<1,and let B={B_(t),t≥0}be an independent Brownian motion.In this study,we investigate the parameter estimation of a mixed fractional Black-Scholes modelS_(t)^(H)=S_(0)^(H)+μ∫_(0)^(t)S_(s)^(H)ds+σ∫_(0)^(t)S_(s)^(H)d(B_(s)+B_(s)^(H)),whereσ>0,μ∈Rare two unknown parameters.Using quasi-likelihood estimation,when the system is observed at some discrete time instants{t_(i)=ih,i=0,1,2,…,n},we give estimations of the parametersμandσprovided h=h(n)→0 nh→∞and h^(1+γ)n→1for someγ>0,as n→∞.We present the asymptotic normality of the estimators based on the velocity of nh^(1+γ)-1 tending to zero as n tends to infinity.Finally,we perform numerical calculus and simulations using factual data from the stock market to verify the effectiveness of the established estimators.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11971101).
文摘Let BH={BHt,t≥0}be a fractional Brownian motion with Hurst index H∈(0,1).Inspired by pathwise integrals and Wick product,in this paper,we consider the forward and symmetric Wick-Itôintegrals with respect to BH as follows:∫t0us⋄d−BHs=limε↓01ε∫t0us⋄(BHs+ε−BHs)ds,∫t0us⋄d∘BHs=limε↓012ε∫t0us⋄(BHs+ε−BH(s−ε)∨0)ds,in probability,where◊denotes the Wick product.We show that the two integrals coincide with divergence-type integral of BH for all H∈(0,1).
基金Acknowledgements The authors would like to thank the anonymous referees whose remarks and suggestions greatly improved the presentation of the paper. Guangjun Shen was supported in part by the National Natural Science Foundation of China （Grant No. 11271020） and the Natural Science Foundation of Anhui Province （1208085MA11）. Litan YAN was partially supported by the National Natural Science Foundation of China （Grant No. 11171062） and the Innovation Program of Shanghai Municipal Education Commission （12ZZ063）.
文摘Let B^H1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequence Sn：=∑i=0^n-1K（n^αBi^H,K1）（Bi＋1^H2,K2-Bi^H2,K2）where K is a standard Gaussian kernel function and the bandwidth parameter α satisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion B^H1,K1. We also give the stable convergence of the sequence Sn by using the techniques of the Malliavin calculus.
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.
基金supported by the National Natural Science Foundation of China(Grant No.11971101).
文摘Let B^(H)be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the self-repelling diffusion X_(t)^(H)=B_(t)^(H)+∫_(0)^(t)∫_(0)^(s)g(u)dX_(u)^(H)ds+vt.wherc v∈R,and g is a nonncgative Borel function.The process is an analogue of lincar self-interacting diffusion(M.Cranston and Y.Le Jan Math.Ann.303(1995).87-93.).Based on the asymptotic behavior of the weight function g at infinity,we establish the large time behavior of the recursive convergence of the solution xH.For example,when g∈C^(∞)(R_(+))and 0<g(t)→+∞(t→+∞)。we demonstrate that there is a sequcuce{λ_(n)}of positive real numbers such that J_(t)^(H)(0:g):=g(t)c^(-G(t))X_(t)^(H)→ν+ξ_(∞)^(H)and J_(t)^(H)(0:g):=G(t)(J_(t)^(H)(n-1:g)-λ_(n-1)(cs_(∞)^(H)+ν))→λ_(n)(cs_(∞)^(H)+ν)(t→+∞)in L^(2)and almost surely for every n∈{1,2....}.where G(t)=∫_(0)^(t)g(s)ds and cS_(∞)^(H):=∫_(0)^(∞)g(r)c^(-G(r))dB_(r)^(H).
