A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

Let u be a positive solution of the ultraparabolic equation $$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$ where 1 ≤ k ≤ n and 0 < T ≤ + ∞. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) ? H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that \(\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0\) , where l = log u, and \(\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2} \) .     

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in ? ^d and a one-dimensional Brownian motion in ?^?+?. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L ^p (? ^d ) function.     

分式Brown运动的极函数

设X(t)(t∈R^N)是指数为α的d维分式Brown运动。本文研究X（t）的极函数问题，得出了满足P｛t∈R^N\｛0｝，X（t）=f(t)=0｝的连续函数f组成的类的特征，解决了Legall提出的一个问题；并且得到了（N，N，2α）过程的不动点的Hausdorff维数。     

利用Ito公式求布朗运动和几何布朗运动的矩

利用Ito公式及Ito积分的性质求出了布朗运动和几何布朗运动的矩的一般形式,同时指出可以利用这种方法求其他扩散过程的矩.     

On Changes of Measure for Super Brownian Motion

Let be a superprocess under the measure P. We show the existence of probability measures which are absolutely continuous with respect to P, and whose Randon–Nikodym derivatives are suitably normalized functions of the self intersection local time of . These measures correspond to measure valued processes exhibiting a certain amount of (reinforcing) self interaction in terms of both the particle motion and the branching mechanism.     

临界超布朗运动的渐近行为

本文证明了在一般的初始测度下，Dawson所获得的临界超布朗运动的渐 近定理并不成立，对于一般分支特征的临界的超布朗运动以及一般寝始测度，本文获得了它的渐近行为的阶的两个估计。     

分数布朗运动参数估计

为了估计分形布朗运动参数,作者进一步修改了Wornell算法.克服了Wornell算法要求样本数量大的缺陷.在仿真中,和lance等人修改的Wornell算法相比较,算法达到了更高的精度,并以均方根误差和标准差为指标说明了本文方法的优越性.     

超布朗运动在临近灭绝时的行为

本文研究具有一类较广泛分支特征的超布朗运动在临近灭绝时的行为，主要得到了在临近灭绝时它的支撑的直径几乎处处收敛到零     

锥中一类调和函数的增长估计

给出了锥中一类调和函数在无穷远点处的增长估计,推广了Siegel和Talvila,张和邓在半空间的相关结果.     

On Upper Class of Increments of Fractional Brownian Motion

A series test for the upper upper class and lower upper class of increments of fractional brownian motion is provided.     

角域上的调和函数与条件Brown运动

设中的角域．该文绘出了A上正调和函数的Martin表示，讨论了极小调和函数与条件Brown运动的一个0—1律之间的关系，并给出了A上极小调和函数的表现形式，     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

Lower Bounds for the Distribution of Suprema of Brownian Increments and Brownian Motion Normalized by the Corresponding Modulus Functions

The Lévy–Ciesielski construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process \((W_{t})_{t\in \left[ 0,T\right] }\) normalized by the global modulus function, for all positive \(\varepsilon \) and \(\delta \). Additionally, uniform results over \(\delta \) are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.     

First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Let (B _t) _t0 be standard Brownian motion starting at y, X _t = x + ^t ₀ V(B _s) ds for x (a, b), with V(y) = y if y0, V(y)=–K(–y) if y0, where >0 and K is a given positive constant. Set _ab=inf{t>0: X _t(a, b)} and ₀=inf{t>0: B _t=0}. In this paper we give several informations about the random variable _ab. We namely evaluate the moments of the random variables , and also show how to calculate the expectations . Then, we explicitly determine the probability laws of the random variables as well as the probability by means of special functions.     

一类二维非可分细分函数的正则性估计

设Ｍ-细分函数用于对应于二维图像处理中梅花状子取样的小波基的构造．本文用迹类算子的谱半径给出了具有指数衰减面具的Ｍ-细分函数Ｓｏｂｏｌｅｖ-指数的一个估计.     

Harnack inequality and derivative formula for SDE driven by fractional Brownian motion

In the paper, Harnack inequality and derivative formula are established for stochastic differential equation driven by fractional Brownian motion with Hurst parameter H < 1/2. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.     

An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related stationary operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.     

Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion

The purpose of this paper is to present a survey on Yor's formula on the probability densities of the exponential functionals represented as integrals in time of geometric Brownian motions and to present results on numerical computations for the densities. We perform the computations via another formula for the densities obtained by Dufresne and we show numerically the desired coincidence in some cases. As an application, we compute the price of an Asian option. AMS 2000 Subject Classification: 65C50, 60J65