Asymptotic Theory for Fractional Regression Models via Malliavin Calculus

We study the asymptotic behavior as n→∞ of the sequence

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.     

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.     

Existence of Solution for Delay Fractional Differential Equations

This paper is devoted to the existence of solution for a class of delay fractional differential equations. First we prove some singular integral inequalities. Then using a fixed point theorem of nonlinear alternative Leray–Schauder type, the existence of at least one solution is proved. To illustrate the main result some examples are given.     

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.     

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.     

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.     

Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.     

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.     

Existence Results for Fractional Order Semilinear Integro-Differential Evolution Equations with Infinite Delay

This paper is concerned with existence results of mild solutions for fractional order semilinear integro-differential evolution equations (FSIDEEs) and semilinear neutral integro-differential evolution equations (FSNIDEEs in short) with infinite delay in α-norm. Our tools include the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and the Krasnoselskii–Schaefer type fixed point theorem.     

Solvability for Impulsive Neutral Integro-Differential Equations with State-Dependent Delay via Fractional Operators

This paper is mainly concerned with the existence of mild solutions for a first-order impulsive neutral integro-differential equation with state-dependent delay. We assume that the undelayed part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed-point theorem for condensing maps combined with theories of analytic resolvent operators, we prove some existence theorems. As an application of these main theorems, some practical consequences are derived.     

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     

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.     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.     

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.     

Existence of Mild Solutions for a Class of Fractional Non-autonomous Evolution Equations with Delay

In this paper, we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo’s fractional derivatives. By using the measure of noncompactness, β-resolvent family, fixed point theorems and Banach contraction mapping principle, we improve and generalizes some related results on this topic. At last, we give an example to illustrate the application of the main results of this paper.