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Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces
Cui, Jing,Yan, Litan,Wu, Xiaotai 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.3
In this paper, we study the existence results of mild solutions for a class of stochastic integro-differential equations with nonlocal conditions and stochastic impulsive integro-differential equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for the existence of mild solutions are derived by means of Leray-Schauder nonlinear alternative. An example is provided to illustrate the theory.
Remarks on asymptotic behavior of weighted quadratic variation of subfractional Brownian motion
Junfeng Liu,Litan Yan 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.2
The present note is devoted to prove, by means of Malliavin calculus, the convergence in L2 of some properly renormalized weighted quadratic variation of sub-fractional Brownian motion SH with parameter H < 1/4.
On the convergence to the multiple subfractional Wiener–Itô integral
Guangjun Shen,Litan Yan,Chao Chen 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.4
In this paper, we construct a family of continuous stochastic processes that converges in law to the multiple Wiener–Itô integrals with respect to the subfractional Brownian motion withH > 12 for the integrand f in a rather general class of functions.Wemainly use Donsker and Stroock approximations and the techniques of the multiple Wiener–Itô integral with respect to the Wiener process.
Central limit theorem for weighted local time of L2 modulus of fractional Brownian motion
Chao Chen,Litan Yan,Cheng Ju 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.4
In this paper, we mainly prove a central limit theorem for weighted local time of L2-modulus of fractional Brownian motion with Hurst parameter H ∈ (12, 1). Similar to Hu and Nualart (2009), based on techniques of stochastic analysis, the main ingredients of the proof are analogous to the asymptotic version of Knight’s theorem and the fractional Clark–Ocone formula for the L2-modulus of the weighted local time increments.
On the convergence to the multiple subfractional Wiener-It$\hat{o}$ integral
Shen, Guangjun,Yan, Litan,Chen, Chao 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.4
In this paper, we construct a family of continuous stochastic processes that converges in law to the multiple Wiener-It$\hat{o}$ integrals with respect to the subfractional Brownian motion with H > $\frac{1}{2}$ for the integrand f in a rather general class of functions. Wemainly use Donsker and Stroock approximations and the techniques of the multiple Wiener-It$\hat{o}$ integral with respect to the Wiener process.
Central limit theorem for weighted local time of $L^2$ modulus of fractional Brownian motion
Chen, Chao,Yan, Litan,Ju, Cheng 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.4
In this paper, we mainly prove a central limit theorem for weighted local time of $L^2$-modulus of fractional Brownian motion with Hurst parameter $H{\in}(\frac{1}{2},\;1)$. Similar to Hu and Nualart (2009), based on techniques of stochastic analysis, the main ingredients of the proof are analogous to the asymptotic version of Knight's theorem and the fractional Clark-Ocone formula for the $L^2$-modulus of the weighted local time increments.
Remarks on an integral functional driven by sub-fractional Brownian motion
Guangjun Shen,Litan Yan 한국통계학회 2011 Journal of the Korean Statistical Society Vol.40 No.3
This paper studies the functionals A1(t, x) = ∫ ^t _01_[0,∞)(x − S^H _s )ds,A_2(t, x) = ∫^ t _01_[0,∞)(x − S^H _s )s^(2H−1)ds,where (S^H _t )0≤t≤T is a one-dimension sub-fractional Brownian motion with index H ∈ (0, 1). It shows that there exists a constant pH ∈ (1, 2) such that p-variation of the process A_j(t, S^H_t ) − ^t _0 L_j(s, ^S_H s )dS^H _s (j = 1, 2) is equal to 0 if p > pH, where L_j, j = 1, 2, are the local time and weighted local time of S^H, respectively. This extends the classical results for Brownian motion.
Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces
Jing Cui,Litan Yan,Xiaotai Wua 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.3
In this paper, we study the existence results of mild solutions for a class of stochastic integro-differential equations with nonlocal conditions and stochastic impulsive integrodifferential equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for the existence of mild solutions are derived by means of Leray–Schauder nonlinear alternative. An example is provided to illustrate the theory.