arXiv subjects
Publications and source records attributed to Yaozhong Hu.
We extend the classical Garsia-Rodemich-Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint Hölder continuity along the rectangles for fractional Brownian fields $W(t, x)$ and for the solution $u(t, y)$ of stochastic heat equation with additive white noise.
In this paper, we establish a version of the Feynman-Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman-Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogenous Gaussian noise: First, it is obtained an explicit expression for the Malliavin derivatives of the solutions. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.
In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter $H<1/2$. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman--Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic partial differential equation.
We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter $H>1/2$). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motion and the corresponding underlying standard Brownian motion. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motion while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.
In this paper we compute the $\frac 43$-variation of the derivative of the self-intersection Brownian local time $γ_t=\int_0^t \int_0^u δ'(B_u-B_s)dsdu\,, t\ge 0$, applying techniques from the theory of fractional martingales.
In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the $L^p$-Hölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained $L^p$-Hölder continuity results. The main tool is the Malliavin calculus.
The Hölder continuity of the solution to a nonlinear stochastic partial differential equation arising from one dimensional super process is obtained. It is proved that the Hölder exponent in time variable is as close as to 1/4, improving the result of 1/10 in a recent paper by Li et al [3]. The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.
In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue's measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the rough paths context.
In this article, we derive a Stratonovich and Skorohod type change of variables formula for a multidimensional Gaussian process with low Hölder regularity (typically lower than 1/4). To this aim, we combine tools from rough paths theory and stochastic analysis.
We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.
The purpose of this note is to prove a central limit theorem for the $L^3$-modulus of continuity of the Brownian local time using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus of the Brownian local time.
In this paper we introduce the notion of fractional martingale as the fractional derivative of order $α$ of a continuous local martingale, where $α\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2α}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy's characterization theorem for the fractional Brownian motion.
The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.
This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the $S$-transform which is also described without the introduction of generalized random variables.
We study a least squares estimator $\hat θ_T$ for the Ornstein-Uhlenbeck process, $dX_t=θX_t dt+σdB^H_t$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac12$. We prove the strong consistence of $\hat θ_T$ (the almost surely convergence of $\hat θ_T$ to the true parameter ${% θ}$). We also obtain the rate of this convergence when $1/2\le H<3/4$, applying a central limit theorem for multiple Wiener integrals. This least squares estimator can be used to study other more simulation friendly estimators such as the estimator $\tilde θ_T$ defined by (4.1).
In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.
In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time $t>0$.
The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos development for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.