Paul Gassiat, Aleksandar Mijatović, Harald Oberhauser
We derive a nonlinear integral equation to calculate Root's solution of the Skorokhod embedding problem for atom-free target measures. We then use this to efficiently generate bounded time-space increments of Brownian motion and give a parabolic version of Muller's classic "Random walk over spheres" algorithm.
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In this short article, we will focus on the different links between some stochastic processes resulting from Brownian motion and two notions of probability theory (proportional increments and last hitting times).
We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain properties of covariance function for the corresponding stationary fields. We present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.
In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments. We give sufficient conditions for the characteristic function of the process with independent increments that ensure the existence of the approximation. We apply these results to Lévy processes. Finally we extend this results to the $m$-dimensional complex Brownian motion.
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 $δ$. Additionally, uniform results over $δ$ 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…
On the increments of the principal value of Brownian local time
设$W$是从0开始的一维布朗运动。定义$Y(t)= \int_0^t{\d s \over W(s)} := \lim_{ε\to0} \int_0^t 1_{(|W(s)|> ε)} {\d s \over W(s)} $为与当地时间相关的柯西主值。我们证明了$Y$增量的Limsup和Liminf结果。
A random walk approximation to fractional Brownian motion
我们提出了分数布朗运动的随机游走逼近，其中分数随机游动的增量被定义为伯努利随机游动过去增量的加权和。
Our purpose is to investigate properties for processes with stationary and independent increments under $G$-expectation. As applications, we prove the martingale characterization to $G$-Brownian motion and present a decomposition for generalized $G$-Brownian motion.
Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$
研究在单位正方形内由泊松点过程生成的点集，基于德劳内三角剖分考虑归一化增量和，推导当点密度趋于无穷时的中心极限定理。
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (Séminaire de Probabilités XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exc…
Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}η$$ as $h\to 0$, where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
研究两个独立同分布各向同性分数布朗运动场在固定二维域上的点wise最大值的平方增量和，揭示其在Hurst参数H<1/2时的渐近行为，证明归一化后收敛于确定常数乘以两个场差的局部时间。
The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken as the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownian motion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting of independent coefficients for $H = 1/2$. We obtain that the average number of the real zeros of $P_n(x)$ is~$\sim K_H \log n$, for large $n$, where $K_H = (1 + 2 \sqrt{H(1-H)})/π$ (a generalisation of a classical re…
We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright-Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017) [JS17]. The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.
In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.
We investigate here the Central Limit Theorem of the Increment Ratio Statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer-Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments.…
Let $W_t$ be a standard Brownian motion. It is well-known that the Langevin equation $d U_t = -θU_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion $W_t$ with some other process $G$ with stationary increments. In this article we prove that the converse also holds and all continuous stationary processes arise from a Langevin equation with certain noise $G=G_θ$. Discrete analogies of our results are given and applications are discussed.
Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data
研究KPZ固定点在任意初始数据下的空间增量与速率二布朗运动的强定量比较，证明其在紧集上的一致性，并应用得到空间增量的一侧大偏差不等式及中心KPZ固定点的韦纳密度熵有限性。
We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of inc…
Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.
We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a care…
Renewal series and square-root boundaries for Bessel processes
我们展示了如何描述布朗指数泛函作为一个更新系列提供了访问的法律的首中时的平方根边界的贝塞尔过程。这扩展了Breiman和Shepp关于布朗运动的经典结果，并通过不同的手段恢复了Bessel过程的扩展，由Delong和Yor独立获得。
An algorithm for simulating Brownian increments on a sphere
本文提出球面上布朗运动转移密度的新公式，并基于该公式与Wright-Fisher扩散的对偶性，开发了模拟球面布朗运动增量的算法，数值分析表明时间步长不太小时算法稳定。
The 1-d stochastic wave equation driven by a fractional Brownian motion
在本文中，我们开发了一个年轻的积分理论在2维，这将使我们能够解决一个非线性一维波动方程驱动的任意信号的矩形增量满足一些Hölder正则性条件，对于一些Hölder指数大于1/2。这一结果将应用于无穷维分数布朗运动。
Error Distribution of the Local Linearization Method for Stochastic Differential Equations with Additive Brownian Noise
该研究证明加性布朗噪声随机微分方程局部线性化格式离散误差的泛函稳定极限定理，明确误差结构与归一化因子，给出收敛条件及极限过程的驱动项与系数构成。
We study properties of the random metric space called the Brownian map. For every h>0, we consider the connected components of the complement of the open ball of radius h centered at the root, and we let N(h,r) be the number of those connected components that intersect the complement of the ball of radius h+r. We then prove that r^3N(h,r) converges as r tends to 0 to a constant times the density at h of the profile of distances from the root. In terms of the Brownian cactus, this gives asymptotics for the number of vertices at height h that have descendants at height h+r. Our proofs are based …
Strong approximations of uniform transport processes to the standard Brownian motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego et al. (1971)) and improves their rate of strong convergence (Gorostiza et al. (1980)). We finalise by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
The generalized grey Brownian motion is a time continuous self-similar with stationary increments stochastic process whose one dimensional distributions are the fundamental solutions of a stretched time fractional differential equation. Moreover, the distribution of the time-changed Brownian motion by an inverse stable process solves the same equation, hence both processes have the same one dimensional distribution. In this paper we show the mutual singularity of the probability measures on the path space which are induced by generalized grey Brownian motion and the time-changed Brownian motio…
The scaling limit of the root component in the Wired Minimal Spanning Forest of the Poisson Weighted Infinite Tree
本文证明了泊松加权无限树的有线最小生成森林中根连通分量的标度极限，该极限由独立布朗树通过两种粘合过程聚合而成。
A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics
在本文中，我们提出了一个一般的数学结构，使我们能够定义一个参数类的$H$ -sssi随机过程（自相似的固定增量），它具有边际概率密度函数，根据分数型的部分积分微分方程的时间演变。这种构造是基于函数空间上的有限测度理论。由于方差随时间演化为幂函数，因此这些$H$ -sssi过程自然地提供了慢和快异常扩散的模型。作为特殊情况，这类运动包括分数布朗运动、灰色布朗运动和布朗运动。
This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and spectral analyses of the increments are investigated. On the other hand we show that (almost) all mfBm's may be reached as the limit of partial sums of (super)linear processes. Finally, an algorithm to perfectly simulate the mfBm is presented and illustrated by some simulations.
We consider a stochastic process $Y$ defined by an integral in quadratic mean of a deterministic function $f$ with respect to a Gaussian process $X$, which need not have stationary increments. For a class of Gaussian processes $X$, it is proved that sums of properly weighted powers of increments of $Y$ over a sequence of partitions of a time interval converge almost surely. The conditions of this result are expressed in terms of the $p$-variation of the covariance function of $X$. In particular, the result holds when $X$ is a fractional Brownian motion, a subfractional Brownian motion and a bi…
Geodesics in large planar maps and in the Brownian map
我们研究随机度量空间中的测地线，称之为布朗映射，它表现为大平面映射的标度极限。特别是，我们完全描述测地线从称为根的区别点开始，我们的特点是由一个以上的测地线连接到根的所有点的集合S。集合S在布朗映射中是稠密的，并且同胚于一个非紧实树。此外，对于S中的每个x，从x到根的不同测地线的数量等于S中{x}的补的连通分量的数量。特别地，布朗映射的点可以通过至多三条不同的测地线连接到根。我们的结果有应用程序的大地线在大平面地图的行为。
In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.
We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.
Let $(\mathcal{T},d)$ be the random real tree with root $ρ$ coded by a Brownian excursion. So $(\mathcal{T},d)$ is (up to normalisation) Aldous CRT \cite{AldousI} (see Le Gall \cite{LG91}). The $a$-level set of $\mathcal{T}$ is the set $\mathcal{T}(a)$ of all points in $\mathcal{T}$ that are at distance $a$ from the root. We know from Duquesne and Le Gall \cite{DuLG06} that for any fixed $a\in (0, \infty)$, the measure $\ell^a$ that is induced on $\mathcal{T}(a)$ by the local time at $a$ of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in $\mathcal…
This paper develops a generalization of Brownian motion with stationary, autocorrelated increments as a tractable model for problems in business and finance. We show that any real continuous Gaussian Markov process with stationary increments and smooth covariance function is characterized by three parameters quantifying drift, volatility, and autocorrelations. We model a queue as a functional of a process defined by those characteristics and derive its transient distribution conditional on its history.
For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_γ(t)=X(t)-ct-γ\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $γ$-reflected process, where $γ\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex f…
For each $n\in\mathbb{N}$, let $\mathbf{Q}_n$ be a uniform rooted measured quadrangulation of size $n$ conditioned to have $r(n)$ vertices in its root block. We prove that for a suitable function $r(n)$, after rescaling graph distance by $\left(\frac{21}{40\cdot r(n)}\right)^{1/4}$, with an appropriate rescaling of measure, $\mathbf{Q}_n$ converges to a random pointed measured non-compact metric space $\boldsymbol{ \mathcal{S}}$, in the local Gromov-Hausdorff-Prokhorov topology; the space $\boldsymbol{ \mathcal{S}}$ is built by identifying a uniform point of the Brownian map with the distingui…
Small Brownian Loops Hitting SLE$_2$: An Exact Natural-Content Limit
该研究在有界解析若尔当域中证明了击中弦形$\text{SLE}_2$的小布朗环的根强度测度经$ε^{5/4}$缩放后依$L^1$收敛到自然含量测度的常数倍，给出精确极限常数，并得到淡收敛结果与布朗环汤的大数定律应用。
We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the quadratic variations of the solution, defined via higher order increments. Then we apply our results to construct and study estimators for the Hurst index.
Approximation of Random Differential Equations Driven by Physical Brownian Motion with Fast Oscillating Noise
研究由满足奇异摄动朗之万方程且具有缩放混合随机力的半鞅驱动的随机微分方程逼近。用扩散逼近法探索半鞅粗糙路径提升的极限，应用通用极限定理确定随机微分方程极限，还建立了混合随机力的迭代弱不变原理。
Operator fractional Brownian fields (OFBFs) are Gaussian, stationary-increment vector random fields that satisfy the operator self-similarity relation {X(c^{E}t)}_{t in R^m} L= {c^{H}X(t)}_{t in R^m}. We establish a general harmonizable representation (Fourier domain stochastic integral) for OFBFs. Under additional assumptions, we also show how the harmonizable representation can be reexpressed as a moving average stochastic integral, thus answering an open problem described in Bierme et al.(2007), "Operator scaling stable random fields", Stochastic Processes and their Applications 117, 312--3…
The goal of this paper is to establish a relation between characteristic polynomials of $N\times N$ GUE random matrices $\mathcal{H}$ as $N\to\infty$, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of $D_N(z)=-\log|\det(\mathcal{H}-zI)|$ on mesoscopic scales as $N\to\infty$. By employing a Fourier integral representation, we use this to prove a continuous anal…
Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
本文针对离散观测的漂移多子分数布朗运动，证明了完整有限样本似然理论的存在，给出显式极大似然估计量、精确水平的区间与检验，并验证了强相合性、渐近正态性及模拟效果。
Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$
研究\(\mathrm{GL}(N,\mathbb{C})\)上布朗运动特征值，通过一族非退化扩散过程及对小\(t\)时布朗运动的定量逼近，证明其特征值经验法则几乎必然收敛到确定性概率测度，解决了相关猜想。
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