共查询到20条相似文献,搜索用时 15 毫秒
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We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such integral can have any prescribed distribution. We discuss some applications of these results, in particular, to fractional Black–Scholes model of financial market. 相似文献
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We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a -dimensional fractional Brownian motion (fBm) with Hurst parameter , where the integrands are vector fields applied to . It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion. 相似文献
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Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product
integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved
as an application of multiple fractional integrals.
Received: 22 September 1998 / Revised version: 20 April 1999 相似文献
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Salim Noreddine 《Journal of multivariate analysis》2011,102(6):1008-1017
By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals Fn towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of Fn to zero; (ii) the covariance matrix of Fn to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a Rd-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor. 相似文献
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Multiple fractional integrals 总被引:2,自引:0,他引:2
Multiple integrals with respect to fractional Brownian motion (with H > 1/2) are constructed for a large class of functions. The first and second moments of the multiple integrals are explicitly
identified.
Received: 23 February 1998 / Revised version: 31 July 1998 相似文献
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Fourier normal ordering (Unterberger, 2009) [34] is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion B with arbitrary Hurst index α (in particular, for α≤1/4, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of B defined in Unterberger (2009) [32]. The regularization procedure is applied to ‘Fourier normal ordered’ iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees (Connes and Kreimer, 1998) [6] for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments. 相似文献
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We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application. 相似文献
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L.H. Duc M.J. Garrido-Atienza A. Neuenkirch B. Schmalfuß 《Journal of Differential Equations》2018,264(2):1119-1145
This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion with covariance operator Q, provided that and is sufficiently small. 相似文献
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Oana Mocioalca 《Journal of Functional Analysis》2005,222(2):385-434
We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given. 相似文献
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In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet. 相似文献
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Summary In this paper we study some classes of Wiener functionals whose elements can be composed with a non-linear, non-absolutely continous transformation of the form of perturbation of identity in the direction of Cameron-Martin space. We show that under certain conditions the image of the Wiener measure under the above transformation induces a generalized Wiener functional on certain Sobolev spaces generalizing the Radon-Nikodym relation to non absolutely continuous transformations. A series representation for the generalized Radon-Nikodym derivative is presented and conditional expectations of some generalized random variables are considered. 相似文献
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Chao ChenLitan Yan 《Statistics & probability letters》2011,81(8):1003-1012
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Summary LetL(x, T),xR
d
,TR
+
N
, be the local time of theN-parameter Wiener processW taking values inR
d
. Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals
asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X
T
(f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X
T
(f)–E(X
T
(f)).This research was made during a stay at the LMU in München supported by DAAD 相似文献
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We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index K into fractional Brownian motion of index H. Integration is carried out over [0,t], t>0. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L2(P)-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t], t>0. 相似文献
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Summary Let be a centered Gaussian measure on a Hilbert spaceH and let
be the centered ball of radiusR>0. ForaH and
, we give the exact asymptotics of (B
R(t)+t·a) ast. Also, upper and lower bounds are given when is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.Supported in part by NSF grant number DMS-9024961 相似文献
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A characterization of a regular family of semimatingales as a maximal fasmily of processes with respect of which one can define a stochastic line integral with natural continuity properties is given. 相似文献