Yaozhong Hu David Nualart
We derive estimates for the solutions to differential equations driven by a Hölder continuous function of order $β>1/2$. As an application we deduce the existence of moments for the solutions to stochastic partial differential equations driven by a fractional Brownian motion with Hurst parameter $H>{1/2}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a H{ö}lder drift, for a H{ö}lder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin, Veretennikov, Krylov and R{ö}ckner from non-degenerate to degenerate cases. The existence of a threshold for the H{ö}lder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is b…
In this paper, we prove weak uniqueness of hypoelliptic stochastic differential equation with H{ö}lder drift, with H{ö}lder exponent strictly greater than 1/3. We then extend to a weak framework the previous work [CdR12] where strong uniqueness was proved when the H{ö}lder exponent is strictly greater than 2/3. We also show that this result is sharp, by giving a counter example to weak uniqueness when the H{ö}lder exponent is just below 1/3. Our approach is based on martingale problem formulation of Stroock and Varadhan and is based on smoothing properties of the associated PDE.
The weak averaging principle of stochastic functional partial differential equations with H$\ddot{\text{o}}$lder continuous coefficients and infinite delay
本文针对具有Hölder连续系数和无限时滞的随机泛函偏微分方程，提出广义耦合方法，证明弱解存在唯一性并建立弱平均原理，即原系统解在有限区间上依分布收敛到平均系统解，并给出两个应用实例。
In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin's transformation \cite{zvonkin\_transformation\_1974} and so on the regularization properties of the associated PDE, which is stated on the space $[0,T]\times \R^d\times \mathcal{P}\_2(\R^d)$, where $T$ is a positive number, $d$ denotes the dimension equation and $\mathcal{P}\_2(\R^d)$ is the space of probability measures on $\R^d$ with finite second order moment. Especi…
This paper investigates existence results for path-dependent differential equations driven by a H{ö}lder function where the integrals are understood in the Young sense. The two main results are proved via an application of Schauder theorem and the vector field is allowed to be unbounded. The H{ö}lder function is typically the trajectory of a stochastic process.
In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal condition. Singularity means that at the final time, the value of the solution can be equal to infinity. Different types of regularity of this viscosity solution are investigated: Sobolev, H{ö}lder or strong regularity.
The It{ô} map assigns the solution of a Rough Differential Equation, a generalization of an Ordinary Differential Equation driven by an irregular path, when existence and uniqueness hold. By studying how a path is transformed through the vector field which is integrated, we prove that the It{ô} map is H{ö}lder or Lipschitz continuous with respect to all its parameters. This result unifies and weaken the hypotheses of the regularity results already established in the literature.
Weak error for SDEs with additive stable noise and singular drift: choose the test function in the same space as the drift!
本文研究了具有加性稳定噪声和奇异漂移的SDE的弱误差，强调在相同空间中选择测试函数以提高收敛率。
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{ö}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes th…
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild H{ö}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $α\in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.
This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with Hölder regularity greater than 1/4. After recalling how to treat the case of ordinary stochastic differential equations, we mainly focus on the case of delay equations. A careful analysis is then performed in order to show that a fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions of our abstract theorems.
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
Quantitative weak propagation of chaos for stable-driven McKean-Vlasov SDEs
本文研究旋转不变α-稳定过程驱动的McKean-Vlasov随机微分方程，利用对应半群的正则性与倒向Kolmogorov方程描述的动力学，在半群层面证明了平均场相互作用粒子系统的新混沌传播估计。
The Navier-Stokes equation on Rd (d greater or equal to 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_ r, with r > 1 + d is obtained. We also show p,p p the convergence to solutions of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_ pr,q, with p > 1, 1 greater or equal to q greater or equal to infinity, r > max(1, d); here the maximal time interval depends on p the viscosity.
We study the anticipative backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H greater than 1/2. The stochastic integral used throughout the paper is the divergence operator type integral. We obtain the existence and uniqueness of solutions to these equations. A comparison theorem for this type of anticipative BSDEs is also established.
In this paper, we focus on the mean-field backward stochastic differential equations (BSDEs) driven by a fractional Brownian motion with Hurst parameter H greater then 1/2. First, the existence and uniqueness of these equations are established under Lipschitz condition. Then, a comparison theorem for such mean-field BSDEs is obtained. Finally, as an application, we connect this mean-field BSDE with a nonlocal partial differential equation (PDE).
We study a least squares estimator for an unknown parameter in the drift coefficient of a path- distribution dependent stochastic differential equation involving a small dispersion parameter epsilon greater than zero. The estimator, based on n discrete time observations of the stochastic differential equation, is shown to be convergent weakly to the true value as epsilon goes to zero and n goes to infinity. This indicates that the least squares estimator obtained is consistent with the true value. Moreover, we obtain the rate of convergence and derive the asymptotic distribution of least squar…
We study here a heat-type differential equation of order n greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess (coinciding with the one governed by the standard, non-fractional, equation) with a time argument T which is itself random. The distribution of T is presented together with some features of the solution (such as analytic expressions for its moments).
The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a $γ$-Hölder continuous process with $γ>1/2$ (e.g. a fractional Brownian motion with Hurst parameter greater than $1/2$). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of Euler approximations for mixed stochastic differential equations. As a side result of independent inter…
We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path $w$ on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of $ρ$-\tmtextit{irregularity} and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function $w$ sampled according to th…
We prove uniqueness in law for a class of parabolic stochastic partial differential equations in an interval driven by a functional A(u) of the temperature u times a space-time white noise. The functional A(u) is Hölder continuous in u of order greater than 1/2. Our method involves looking at an associated system of infinite-dimensional stochastic differential equations and we obtain a uniqueness result for such systems.
In this note, we consider an optimal control problem associated to a differential equation driven by a Hölder continuous function g of index greater than 1/2. We split our study in two cases. If the coefficient of dg\_t does not depend on the control process, we prove an existence theorem for a slightly generalized control problem, that is we obtain a literal extension of the corresponding deterministic situation. If the coefficient of dg\_t depends on the control process, we also prove an existence theorem but we are here obliged to restrict the set of controls to sufficiently regular functio…
We define and study fractional versions of the well-known Gamma subordinator $Γ:=\{Γ(t),$ $t\geq 0\},$ which are obtained by time-changing $% Γ$ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding…
In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder 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 smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.
In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,η(t),Y(t),Z(t))dt-Z(t)δB^{H}(t), \quad t\in[0,T], Y(T)=ξ,.] where $η$ is a stochastic process given by $η(t)=η(0) +\int_{0}^{t}σ(s) δB^{H}(s)$, $t\in[0,T]$, and $B^{H}$ is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, \textit{BDSEs driven by fBm}, SIAM J Control Opt…
Non-asymptotic statistical test of the diffusion coefficient of stochastic differential equations
本文基于离散观测提出随机微分方程扩散系数行列式的非渐近统计检验，在固定观测数和时间步长下控制第一类和第二类错误，并提供了各维度的统计量分布、多重检验程序及功效控制条件。
Second-order fields for stochastic partial differential equations
本文通过Da Prato-Debussche分解刻画一维多项式随机偏微分方程解的二次涨落极限，并讨论高维弱耦合情形下的推广。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。