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Yaozhong Hu

Publications and source records attributed to Yaozhong Hu.

At least 91 records · Page 5Linked to original sources
arXiv2015

Density convergence in the Breuer-Major theorem for Gaussian stationary sequences

Consider a Gaussian stationary sequence with unit variance $X=\{X_k;k\in {\mathbb{N}}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_n=n^{-1/2}\sum^{n-1}_{k=0}f(X_k)$, where $f$ designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of $V_n$ towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of $X$.

math.PR↗
arXiv2015

Nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[ρ(u(t,x))\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d, \] where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: $d<2α+\fracαβ\min(2γ-1,0)$. In some cases, the initial data can be measures. When $β\in (0,1]$, we prove the sample path regularity of the solution.

math.PR↗
arXiv2015

Two-point correlation function and Feynman-Kac formula for the stochastic heat equation

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on $\mathbb{R}$. The bounds for $p$-th moments proved in [3] are simplified. We validate the Feynman-Kac formula for the $p$-point correlation function of the solutions to this equation with measure-valued initial data.

math.PR↗
arXiv2015

On the intermittency front of stochastic heat equation driven by colored noises

We study the propagation of high peaks (intermittency front) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.

math.PR↗
arXiv2015

Stochastic Differential Equation for Brox Diffusion

This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $\mathcal{B}$. It is shown that the Itô-McKean representation associated with any Brownian motion (independent of $W$) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term $\int_0^T \dot W(X(t))dt$, the main idea is to use the concept of local time together with the polygonal approximation $W_π$. Some new results on the local time of Brownian motion needed in our proof are established.

math.PR↗
arXiv2015

Stochastic heat equation with rough dependence in space

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.

math.PR↗
arXiv2015

Nonlinear Young integrals and differential systems in Hölder media

For Hölder continuous functions $W(t,x)$ and $ϕ_t$, we define nonlinear integral $\int_a^b W(dt, ϕ_t)$ in various senses, including Itô-Skorohod and pathwise. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with $\dot ϕ_t=(\partial _tW)(t, ϕ_t)$ is also studied and its applications to the transport equation $\partial _t u(t,x)-\partial _t W(t,x)\nabla u(t,x)=0$ in rough media is given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients $\partial _t u(t,x)+Lu(t,x) +u(t,x)W(t,x)=0$ are given, where $L$ is a second order elliptic differential operator with random coefficients (dependent on $W$). To establish such formula the main difficulty is the exponential integrability of some nonlinear integrals, which is proved to be true under some mild conditions on the covariance of $W$. Along the way, we also obtain an upper bound for increments of stochastic processes on multidimensional rectangles by majorizing measures.

math.PR↗
arXiv2015

Nonlinear Young integrals via fractional calculus

For Hölder continuous functions $W(t,x)$ and $φ_t$, we define nonlinear integral $\int_a^b W(dt, φ_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals.

math.PR↗
arXiv2015

Fractional diffusion in Gaussian noisy environment

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^αu(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^α$ is the fractional derivative of order $α$ with respect to the time variable $t$, $\textit{B}$ is a second order elliptic operator with respect to the space variable $x\in\mathbb{R}^d$, and $W^H$ a fractional Gaussian noise of Hurst parameter $H=(H_1, \cdots, H_d)$. We obtain conditions satisfied by $α$ and $H$ so that the square integrable solution $u$ exists uniquely .

math.PR↗
arXiv2014

Feynman-Kac formula for fractional heat equation driven by fractional white noise

In this paper we obtain a Feynman-Kac formula for the solution of a fractional stochastic heat equation driven by fractional noise. One of the main difficulties is to show the exponential integrability of some singular nonlinear functionals of symmetric stable Lévy motion. This difficulty will be overcome by a technique developed in the framework of large deviation. This Feynman-Kac formula is applied to obtain the Hölder continuity and moment formula of the solution.

math.PR↗
arXiv2014

Smoothness of the joint density for spatially homogeneous SPDEs

In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t>0$, assuming some suitable regularity and non degeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.

math.PR↗
arXiv2014

Singular mean-field control games with applications to optimal harvesting and investment problems

This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular mean-field investment game is studied where the Nash equilibrium exists but is not unique.

math.OC↗
arXiv2014

On optimal mean-field type control problems of stochastic systems with jump processes under partial information

This paper considers the problem of partially observed optimal control for forward stochastic systems which are driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When all the system coefficients and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjointprocess is explicitly expressed. We also investigate the mean-field type optimal control problems for systems driven by mean-field type stochastic differential equations (SDEs in short) with jump processes, in which the coefficients contain not only the state process but also its marginal distribution under partially observed information. The maximum principle is established using convex variational technique with an illustrating example about linear-quadratic optimal control.

math.PR↗
arXiv2014

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

math.PR↗
arXiv2013

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein's method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed.

math.PR↗
arXiv2013

On Hölder continuity of the solution of stochastic wave equations

In this paper, we study the stochastic wave equations in the spatial dimension 3 driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path Hölder continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean Hölder continuity of the covariance function or in terms of its spectral measure. Some examples of the covariance functions are proved to satisfy our conditions, which include the case of the work \cite{dalang4}. In particular, we obtain the Hölder continuity results for the solution of the stochastic wave equations driven by (space inhomogeneous) fractional Brownian noises. For this particular noise, the optimality of the obtained Hölder exponents is also discussed.

math.PR↗