arXiv subjects
Publications and source records attributed to Yaozhong Hu.
This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a Hörmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property.
We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \dots, B^{m})$ with Hurst parameter $H \in (\frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, \dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs. Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-\frac12 -H}$. In the general case when $m>1$, the exact rate equals $n^{\frac12 -2H}$. In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions.
We present an explicit solution triplet $(Y, Z, K)$ to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ and $K$ are expressed by Hida-Malliavin derivatives involving $Y$.
In this article, we obtain sharp conditions for the existence of the high order derivatives ($k$-th order) of intersection local time $ \widehatα^{(k)}(0)$ of two independent d-dimensional fractional Brownian motions $B^{H_1}_t$ and $\widetilde{B}^{H_2}_s$ with Hurst parameters $H_1$ and $H_2$, respectively. We also study their exponential integrability.
Gradient and stability type estimates of heat kernel associated with fractional power of a uniformly elliptic operator are obtained. $L^p$-operator norm of semigroups associated with fractional power of two uniformly elliptic operators are also obtained.
This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure convergence of the estimators and the corresponding central limit theorems are obtained for all the Hurst parameter range $H\in (0, 1)$. The least squares estimator is used for the drift parameter. A central limit theorem is proved when the Hurst parameter $H \in (0, 1/2)$ and a noncentral limit theorem is proved for $H\in[3/4, 1)$. Thus, the open problem left in the paper by Hu and Nualart (2010) is completely solved, where a central limit theorem for least squares estimator is proved for $H\in [1/2, 3/4)$.
For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H> \frac12$ it is known that the classical Euler scheme has the rate of convergence $2H-1$. In this paper we introduce a new numerical scheme which is closer to the classical Euler scheme for diffusion processes, in the sense that it has the rate of convergence $2H-\frac12$. In particular, the rate of convergence becomes $\frac 12$ when $H$ is formally set to $\frac 12$ (the rate of Euler scheme for classical Brownian motion). The rate of weak convergence is also deduced for this scheme. The main tools are fractional calculus and Malliavin calculus. We also apply our approach to the classical Euler scheme.
In this paper, we consider a class of backward doubly stochastic differential equations (BDSDE for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the $L^p$-Hölder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the $L^p$-sense. As a by-product, we obtain an explicit representation of the process $Y$ in the solution pair to a linear BDSDE with random coefficients.
We obtain strong consistency and asymptotic normality of a least squares estimator of the drift coefficient for complex-valued Ornstein-Uhlenbeck processes disturbed by fractional noise, extending the result of Y. Hu and D. Nualart, [Statist. Probab. Lett., 80 (2010), 1030-1038] to a special 2-dimensions. The strategy is to exploit the Garsia-Rodemich-Rumsey inequality and complex fourth moment theorems. The main ingredients of this paper are the sample path regularity of a multiple Wiener-Ito integral and two equivalent conditions of complex fourth moment theorems in terms of the contractions of integral kernels and complex Malliavin derivatives.
This paper studies the existence and uniqueness of solution of Itô type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and $dB(t)$ is the Itô differential defined by using Wick product or divergence operator. The coefficients $b$ and $\si$ are random and can be anticipative. Using the relationship between the Itô type and pathwise integrals we first write the equation as a stochastic differential equation involving pathwise integral plus a Malliavin derivative term. To handle this Malliavin derivative term the equation is then further reduced to a system of (two) equations without Malliavin derivative. The reduced system of equations are solved by a careful analysis of Picard iteration, with a new technique to replace the Grönwall lemma which is no longer applicable. The solution of this system of equations is then applied to solve the original Itô type stochastic differential equation up to a positive random time. In the special linear and quasilinear cases the global solutions are proved to exist uniquely.
This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. 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 $n$th moment of the solution.
In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove $u(t,x)\in \mathbb{D}^\infty$ for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux [1] are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.
In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-Δ)^{\fracα{2}}u+u\dot W(t,x)$, where $-(-Δ)^{\fracα{2}}$ with $α\in(0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $α$-stable process. As a byproduct, we obtain the critical values for $θ$ and $η$ such that $\mathbb{E}\exp\left(θ\left(\int_0^1 \int_0^1 |r-s|^{-β_0}γ(X_r-X_s)drds\right)^η\right)$ is finite, where $X$ is $d$-dimensional symmetric $α$-stable process and $γ(x)$ is $|x|^{-β}$ or $\prod_{j=1}^d|x_j|^{-β_j}$.
For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $γ_n^{-1}$, where $γ_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $γ_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $γ_n=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $γ_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.
This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^β+ \fracν{2} (-Δ)^{α/2} \right) u(t,x)= λu(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $β\in (1/2, 2)$, $α\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $H\in(0,1/2)$. We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the $L^p(Ω)$ moments of the solution.
In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the Hölder exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both rates involves with the incomplete Taylor scheme in a very explicit way and then provide us with the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.