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.     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

An enlargement of filtration for Brownian motion

Let Bt be an Ft Brownian motion and Gt be an enlargement of filtration of Ft from some Gaussian random variables. We obtain equations for ht such that Bt ht is a Gt-Brownian motion.     

Iterating Brownian Motions,Ad Libitum

Let B ₁,B ₂,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B _i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W _n (t)=B _n (B _n?1(?(B ₂(B ₁(t)))?)). Although the sequence of processes (W _n ) _n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W _n converge to a random probability measure μ _∞. We then prove that μ _∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W _∞ of independent Brownian motions. We also prove that the collection of random variables (W _∞(t),t∈??{0}) is exchangeable with directing measure μ _∞.     

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     

Régularité Besov-Orlicz du temps local brownien

Let (B_t, t ε [0, 1]) be a linear Brownian motion starting from 0 and denote (L_t(x), t ≥0, x ∈ ℝ) its local time. We prove that, for all t > 0, a.s. the function xL_t (x) belongs to the Besov-Orlicz space B^½M_{2, ∞} with M₂(x)=e^|x|2 -1 and doesn't belong a.s. to B^½,0M_{2, ∞}.     

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).     

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.     

Killed Random Processes and Heat Kernels

Let V(x) ≥ 0 be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion B_t killed at the infinitesimal rate V is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of ∫ ₀ ^t V(B_s)ds. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 423–432, August, 2005. An erratum to this article is available at .     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian H _n of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process {H _[nt]}_0≤t≤1 behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.     

Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII

We penalise Brownian motion by a function of its one-sided supremum considered up to the last zero before t, respectively first zero after t, of that Brownian motion. This study presents some analogy with penalisation by the longest length of Brownian excursions, up to time t.     

Some limit results for probabilities estimates of Brownian motion with polynomial drift

Let (B _t + f(t)) _t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B _t + f(t)) _t∈[0,+∞) are given in this paper.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.     

Ruin probability in the Cramér-Lundberg model with risky investments

We consider the Cramér-Lundberg model with investments in an asset with large volatility, where the premium rate is a bounded nonnegative random function c_t and the price of the invested risk asset follows a geometric Brownian motion with drift a and volatility σ>0. It is proved by Pergamenshchikov and Zeitouny that the probability of ruin, ψ(u), is equal to 1, for any initial endowment u≥0, if ρ?2a/σ²≤1 and the distribution of claim size has an unbounded support. In this paper, we prove that ψ(u)=1 if ρ≤1 without any assumption on the positive claim size.     

Interpolation for partly hidden diffusion processes

Let X_t be n-dimensional diffusion process and S_t be a smooth set-valued function. Suppose X_t is invisible when X_t∈S_t, but we can see the process exactly otherwise. Let X_t0∈S_t0 and we observe the process from the beginning till the signal reappears out of the obstacle after t₀. With this information, we evaluate the estimators for the functionals of X_t on a time interval containing t₀ where the signal is hidden. We solve related 3 PDEs in general cases. We give a generalized last exit decomposition for n-dimensional Brownian motion to evaluate its estimators. An alternative Monte Carlo method is also proposed for Brownian motion. We illustrate several examples and compare the solutions between those by the closed form result, finite difference method, and Monte Carlo simulations.