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1.
In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H1,H2), and (2[0,1],B(2[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in 2[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H1,H2∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.  相似文献   

2.
Analysis of a system of fractional differential equations   总被引:2,自引:0,他引:2  
We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations , where Dα denotes standard Riemann-Liouville fractional derivative, 0<α<1, and A is a square matrix. The unique solution to this initial value problem turns out to be , where Eα denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system , , 0<α<1, and investigate dependence of the solutions on the initial conditions.  相似文献   

3.
Large deviations for local time fractional Brownian motion and applications   总被引:1,自引:0,他引:1  
Let be a fractional Brownian motion of Hurst index H∈(0,1) with values in R, and let be the local time process at zero of a strictly stable Lévy process of index 1<α?2 independent of WH. The α-stable local time fractional Brownian motion is defined by ZH(t)=WH(Lt). The process ZH is self-similar with self-similarity index and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps [P. Becker-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004) 730-756; M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004) 623-638]. However, ZH does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process ZH. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for ZH.  相似文献   

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We construct a family Inεt(f) of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class.  相似文献   

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Let 0<γ<1, b be a BMO function and the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for in the context of the spaces of homogeneous type. The first one establishes that, for A weights, the operator is bounded in the weighted Lp norm by the maximal operator Mγ(Mm), where Mγ is the fractional maximal operator and Mm is the Hardy-Littlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator is bounded from to , where [(m+1)p] is the integer part of (m+1)p and no condition on the weight w is required. From the first inequality we also obtain weighted Lp-Lq estimates for generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.  相似文献   

9.
In this paper we study nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fractional Brownian motion with Hurst parameter and nuclear covariance operator. We establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter H. This result is applied to stochastic parabolic equation perturbed by a fractional white noise. In this case, if the coefficients are Lipschitz continuous and bounded the existence and uniqueness of a solution holds if . The proofs of our results combine techniques of fractional calculus with semigroup estimates.  相似文献   

10.
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