带跳的分数维Brown 运动幂变差的渐近行为

本文研究了X_t = B^H_t + ξ_t 现实幂变差的渐近理论, B^H 为Hurst 指数为H∈(0,1) 的分数维Brown 运动,ξ为与B^H独立的非Gauss Lévy 过程, 我们给出了其大数定律, 以及经适当中心化的中 心极限定理, 这些结果将为处理具有长期记忆跳过程的统计问题提供理论基础.     

Polar functions of multiparameter bifractional Brownian sheets

Let B^H,K ： （B^H,K（t）, t ∈R＋^N} be an （N,d）-bifractional Brownian sheet with Hurst indices H = （H1,..., HN） ∈ （0, 1）^N and K = （K1,..., KN）∈ （0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov＇s entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.     

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index {{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}

Let B?=?(B ₁(t), . . . ,B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index ???<?1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using ??standard?? tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d??emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).     

多参数双分数布朗运动相遇局部时的存在性和联合连续性

本文研究了两个相互独立的(N,d)双分数布朗运动BH1,K1和BH2,K2的相遇局部时的问题.利用Fourier分析,获得了相遇局部时的存在性和联合连续性的结果,推广了分数布朗运动相遇局部时的相关结果.     

Haar-Based Multiresolution Stochastic Processes

Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where ?? _n ?? _n (t) is the integral of the nth Haar wavelet from 0 to t, and ?? _n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X _t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as H?lder continuity and fractional dimensions of graphs of the processes.     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In ?ochowski (2008) [9] we defined truncated variation of Brownian motion with drift, W_t=B_t+μt,t≥0, where (B_t) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c>0. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument.We also define two closely related quantities — upward truncated variation and downward truncated variation.The defined quantities may have interpretations in financial mathematics. The exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process.We calculate the Laplace transform with respect to the time parameter of the moment-generating functions of the upward and downward truncated variations.As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.     

<Emphasis Type="Italic">Φ</Emphasis>-Variation of a bifractional Brownian motion

Let B _H,K = {B _H,K (t)} _t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t _i }_n ⁱ⁼⁰ of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08     

Insider trading equilibrium in a market with memory

We consider the Kyle-Back model for insider trading, with the difference that the classical Brownian motion noise of the noise traders is replaced by the noise of a fractional Brownian motion B ^H with Hurst parameter ${H>\frac{1}{2}}$ (when ${H=\frac{1}{2}, B^H}$ coincides with the classical Brownian motion). Heuristically, for ${H>\frac{1}{2}}$ this means that the noise traders has some ??memory??, in the sense that any increment from time t on has a positive correlation with its value at t. (In other words, the noise trading is a persistent stochastic process). It also means that the paths of the noise trading process are more egular than in the classical Brownian motion case. We obtain an equation for the optimal (relative) trading intensity for the insider in this setting, and we show that when ${H\rightarrow\frac{1}{2}}$ the solution converges to the solution in the classical case. Finally, we discuss how the size of the Hurst coefficient H influences the optimal performance and portfolio of the insider.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

A Wiener test for integrals of brownian motion and the existence of smooth curves in nowhere dense sets

Let J_ω^x(t) = x + ∝₀^tb_ω(s) ds, where b_ω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process J_ω^x(t): Let K be a compact plane set and let x?K. Then if ∑ 2ⁿM₁(A_n(x)?K) < ∞ (where $\text{A}{}_{\mathrm{n}}\text{(x) = {2}{}^{\mathrm{?n?1}}\text{? ¦ z ? x ¦ ? 2}{}^{\mathrm{?n}}\text{}}$ and M₁ denotes one-dimensional Hausdorff content), the process J_ω^x(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes.     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

Grandes déviations du temps local du mouvement brownien fractionnaire

The purpose of this paper is to prove a large deviation principle for a local time of fractional Brownian motion B^H for all H∈(0,1). To cite this article: E.H. Lakhel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 797–801.     

Densité des zéros des transformés de Lévy itérés d'un mouvement brownien

The Lévy transform of a Brownian motion B is the Brownian motion $\text{B′}{}_{\mathrm{t}}\text{=∫}{}_0{}^{\mathrm{t}}\text{sgn}\text{B}{}_{\mathrm{s}}\text{d}\text{B}{}_{\mathrm{s}}$; denote by Bⁿ the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that B_tⁿ=0 for some n, is a.s. dense in the time-axis $\text{R}{}_{\mathrm{+}}$. To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ_t, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫^t₀?(ξ_s) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

The most visited sites of symmetric stable processes

Let X be a symmetric stable process of index α∈ (1,2] and let L ^x _t denote the local time at time t and position x. Let V(t) be such that L _t ^V(t) = sup _{x∈ ℝ} L _t ^x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim _{t →∞} |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion. Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000     

POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS

Let BH,K = {BH,K(t), t ∈ RN+ } be an (N, d)-bifractional Brownian sheet with Hurst indices H=(H1,···,HN) ∈(0,1)N and K=(K1,···,KN) ∈(0,1]N. The properties of the polar sets of BH,K are discussed. The sufficient conditions and necessary conditions for a compact set to be polar for BH,K are proved. The infimum of Hausdorff dimensions of its non-polar sets are obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.