Asymptotic behavior for bi-fractional regression models via Malliavin calculus

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.

INFINITE DIMENSIONAL MALLIAVIN CALCULUS AND ITS APPLICATION

In this paper we study the infinite dimensional Malliavin calculus, and apply it to determingwhen the solution of an infinite dimensional stochastic differential equation has the property thatits finite dimensional distributions possess smooth density.

APPLICATIONS OF MALLIAVIN CALCULUS TO STOCHASTIS DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT COEFFICIENTS

In this paper, we apply Malliavin calculus to discuss when the solutions of stochastic differen-tial equations (SDE's) with time-dependent coefficients have smooth density. Under Hormander'scondition,we conclude that the solutions of the SDE's have smooth density. As a consequence,we get the hypoellipticity for inhomogeneous differential operators.

THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION

Let u(t,x)be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with u(0,x)=1 for all x∈R.In this paper,we prove the law of the iterated logarithm(LIL for short)and the functional LIL for a linear additive functional of the form∫[0,R]u(t,x)dx and the nonlinear additive functionals of the form∫[0,R]g(u(t,x))dx,where g:R→R is nonrandom and Lipschitz continuous,as R→∞for fixed t>0,using the localization argument.

EXACT MAXIMUM LIKELIHOOD ESTIMATOR FOR DRIFT FRACTIONAL BROWNIAN MOTION AT DISCRETE OBSERVATION

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Ess′een bounds for these estimators are obtained by using the Stein’s method via Malliavin calculus.

REMARKS ON SUB-FRACTIONAL BESSEL PROCESSES

Let S = {（St1,···,Std ）}t≥0 denote a d-dimensional sub-fractional Brownian motion with index H ≥ 1/2. In this paper we study some properties of the process X of the formwhere Rt = （（St1）2＋···＋（Std）2）~1/2 is the sub-fractional Bessel process.

PARAMETER ESTIMATION FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A GENERAL GAUSSIAN NOISE

In this paper,we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process(G_(t))t≥0.The second order mixed partial derivative of the covariance function R(t,s)=E[GtGs]can be decomposed into two parts,one of which coincides with that of fractional Brownian motion and the other of which is bounded by(ts)^(β-1)up to a constant factor.This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments;some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion.Under this assumption,we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise(G_(t))t≥0.For the least squares estimator and the second moment estimator constructed from the continuous observations,we prove the strong consistency and the asympotic normality,and obtain the Berry-Esséen bounds.The proof is based on the inner product's representation of the Hilbert space(h)associated with the Gaussian noise(G_(t))t≥0,and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.

D_∞-APPROXIMATION OF PRODUCT VARIATIONS OF TWO PARAMETER SMOOTH SEMIMARTINGALES

Let X be a two parameter smooth semimartingale and (?) be its process of the product variation. It is proved that (?) can be approximated as D_∞-limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the convergence.

SOLUTIONS TO BSDES DRIVEN BY BOTH FRACTIONAL BROWNIAN MOTIONS AND THE UNDERLYING STANDARD BROWNIAN MOTIONS

The local existence and uniqueness of the solutions to backward stochastic differential equations（BSDEs, in short） driven by both fractional Brownian motions with Hurst parameter H ∈ （1/2, 1） and the underlying standard Brownian motions are studied. The generalization of the It6 formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.

WEAK APPROXIMATIONS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL NOISE

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations(SPDEs)driven by additive fractional Brownian motions with the Hurst parameter H∈(1/2,1).The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method.As far as we know,the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature.A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations.In the present work,a novel and efficient approach is presented to carry out the weak error analysis for the approximations,which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus.To the best of our knowledge,the rates of weak convergence,shown to be higher than the strong convergence rates,are revealed in the fractional noise driven SPDE setting for the first time.Numerical examples corroborate the claimed weak orders of convergence.

Mixed Sub-fractional Brownian Motion and Drift Estimation of Related Ornstein-Uhlenbeck Process

In this paper,wewill first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk.In order to verify the rationality of this simulation,we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process,and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter H>1/2.

On the uniqueness result for the BSDE with deterministic coefficient

In this paper,we study one-dimensional backward stochastic differential f equation(BSDE),whose deterministic coefficient is Lipschitz in y but only continuous in.If the terminal conditionξhas bounded Malliavin derivative,we prove some uniqueness results for the BSDE with quadratic and linear growth in,respectively.

Solutions to BSDEs Driven by Both Standard and Fractional Brownian Motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions （or, in short, SFBSDE） axe studied. A Wick-It6 stochastic integral for a fractional Brownian motion is adopted. The fractional It6 formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.

Stochastic Volterra equations driven by fractional Brownian motion

This paper is devoted to study a class of stochastic Volterra equations driven by fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the L^2-metric.

Forward and symmetric Wick-Itôintegrals with respect to fractional Brownian motion

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).

SPDEs with Colored Gaussian Noise: A Survey

This note reviews and discusses some recent results on linear stochastic partial differential equations with multiplicative Gaussian noise colored in time.

Relatively Compact Sets on Abstract Wiener Space

In this note, we obtain a sufficient and necessary condition for a set in an abstract Winner space （X, H, μ） to be relatively compact in L^2（X, μ）. Meanwhile, we give a sufficient condition for relative compactness in L^P（X, μ） for p〉1. We also provide an example of Da Prato-Malliavin Nualart to show the result.

Density functions of doubly-perturbed stochastic differential equations with jumps

We consider a real-valued doubly-perturbed stochastic differential equation driven by a subordinated Brownian motion. By using classic Malliavin calculus, we prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure on R.