Malliavin Calculus for Fractional Delay Equations

In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a H?lder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a C ^??-density. To this purpose, we use Malliavin calculus based on the Fréchet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.     

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.     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

Stochastic Differential Games in Insider Markets via Malliavin Calculus

In this paper, we use techniques of Malliavin calculus and forward integration to present a general stochastic maximum principle for anticipating stochastic differential equations driven by a Lévy type of noise. We apply our result to study a general stochastic differential game problem of an insider.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

Malliavin Calculus for Degenerate Stochastic Functional Differential Equations

Consider the solution {X(t); t∈[?r,T]} of the following stochastic functional differential equation: $$dX(t)=\biggl\{\int_{-r}^{0}\rho(s)X(t+s)\,ds+A_{0}(t,X(t))\biggr\}dt+\sum_{i=1}^{m}A_{i}(t,X(t))\,dW^{i}(t),$$ where ρ(t) is an ?-valued function on [?r,0], and {W(t); t∈[0,T]} is an m-dimensional Brownian motion. The main purpose is to study the smoothness of the probability density of X(T) with respect to the Lebesgue measure.     

Malliavin Calculus for White Noise Driven Parabolic SPDEs

We consider the parabolic SPDE

Tools for Malliavin Calculus in UMD Banach Spaces

In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called umd property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizations, a chain rule for Lipschitz functions, a sufficient condition for pathwise continuity and an Itô formula for non-adapted processes.     

The Itô formula for anticipative processes with nonmonotonous time scale via the Malliavin Calculus

Summary This work is devoted to derive It?-type formulae for anticipative stochastic processes with nonmonotonous time using the Malliavin Calculus techniques and the fundamental theorem of the differential calculus. The same method is applied also to give an It?-Ventcell type formula in the anticipative case.     

On the Malliavin Calculus for SDE's on Hilbert Spaces

We consider a class of SDE's on Hilbert spaces and study the partial hypoellipticity of generators associated with these SDE's. We show that the Malliavin calculus with a new key lemma is efficient for the purpose. The partial Hörmander theorem is proved in this paper, and it is applied to the problem of propagation of absolute continuity of measures by stochastic flows given by those SDE's.     

Information Criteria for Small Diffusions via the Theory of Malliavin–Watanabe

Information criteria based on the expected Kullback–Leibler information are presented by means of the asymptotic expansions derived with the Malliavin calculus. We consider the evaluation problem of statistical models for diffusion processes with small noise. The correction terms are essentially different from the ones for ergodic diffusion models presented in Uchida and Yoshida [34, 35].     

The Ornstein-Uhlenbeck Equation and a Related Malliavin Calculus

We consider two different Brownian motions, B and B ^a ; each of them produces a Wiener-It? chaos representation and therefore it defines a Malliavin derivative, D and D ^a , and a Skorohod integral, δ and δ ^a , respectively. Our aim is to rewrite the differential operators D ^a and δ ^a in terms of D and δ.     

微积分、Malliavin分析与变测度耦合

从微积分中的分部积分公式出发,引入Malliavin分析和变测度耦合方法,并简要介绍它们在随机微分方程研究中的应用,包括建立Bismut公式、Driver公式、Harnack不等式以及推移Harnack不等式.     

Girsanov变换下的Malliavin计算与算子关系

本文讨论了Ｇｉｒｓａｎｏｖ 变换下两个Ｇａｕｓｓ概率空间中Ｍａｌｌｉａｖｉｎ 计算及算子之间的关系     

Asymptotic Theory for Relative-Risk Models with Missing Time-Dependent Covariates

Relative-risk models are often used to characterize the relationship between survival time and time-dependent covariates. When the covariates are observed, the estimation and asymptotic theory for parameters of interest are available; challenges remain when missingness occurs. A popular approach at hand is to jointly model survival data and longitudinal data. This seems efficient, in making use of more information, but the rigorous theoretical studies have long been ignored. For both additive risk models and relative-risk models, we consider the missing data nonignorable. Under general regularity conditions, we prove asymptotic normality for the nonparametric maximum likelihood estimators.     

纵向数据变系数回归模型小波估计的渐近性质

考虑纵向数据下的变系数回归模型y_(ij)=x_(ij)~Tθ(t_(ij))+e_(ij)i=1,2,…,n j=1,2,…,m.利用小波光滑和加权最小二乘方法,分别研究了模型中未知参数θ(·)的小波估计θ(·)和误差方差σ~2的小波估计σ~2,在适当的条件下,证明了θ的强相合性,强相合速度,并得到θ和σ~2的渐近正态性.     

The Fractional Calculus for Some Stochastic Processes

Abstract

The integration and differentiation of fractional orders are well known concepts for deterministic functions (see Miller, K.S.; Ross, B. An Introduction to Fractional Calculus and Fractional Differential Equations; John Wiley: New York, 1993; I. Podlubny and Ahmed M.A. El-Sayed, On two definitions of fractional calculus Slovak Academy of Sciences Institute of experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996; Podlubny, I. Fractional Differential Equations; Acad. Press: San Diego – New York, London etc. 1999; Samko, S.G.; Kilbas, A.A.; Marichev, O. Integral and derivatives of the fractional orders and some of their applications. Nauka i Teknika Minisk 1983). In earlier work, we have studied the fractional calculus for mean square continuous stochastic processes. In this work, we shall study the mean square (m.s.) fractional calculus for stochastic processes which are m.s. Riemann-integrable and prove some its properties.     

经验似然重抽样下回归模型的Bootstrap逼近

本文提出用经验似然重抽样来bootstrap逼近线性回归模型中的学生化最小二乘估计．我们证明了该方法具有一般s－2项Edgeworth展开，它是二阶相合的而且比经典的方法损失更小．     

Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes

We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.