Coalescence of skew Brownian motions

On the time inhomogeneous skew Brownian motion

This paper is devoted to the construction of a solution for the "Inhomogenous skew Brownian motion" equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Étoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.

On the constructions of the skew Brownian motion

This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

Skew-product decomposition of Brownian motion on ellipsoid

In this article we obtain a skew-product decomposition of a Brownian motion on an ellipsoid of dimension $n$ in a Euclidean space of dimension $n+1$. We only consider such ellipsoid whose restriction to first $n$ dimensions is a sphere and its last coordinate depends on a variable parameter. We prove that the projection of this Brownian motion on to the last coordinate is, after a suitable transformation, a Wright-Fisher diffusion process with atypical selection coefficient.

On the existence of a time inhomogeneous skew Brownian motion and some related laws

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation, which first appear in a seminal paper by Sophie Weinryb (1983). We investigate some laws related to the constructed process. In particular, using the description of the straddling excursion above a deterministic time, we compute the joint law of the process, its local time and its straddling time.

平方贝塞尔流线的交汇及其在斜布朗运动中的应用

Meeting of squared Bessel flow lines and application to the skew Brownian motion

本文研究不同维数平方贝塞尔流线的交汇水平，证明其生成跳跃马尔可夫过程，并应用于斜布朗流，扩展局部时流结果，计算分叉时例外时间的豪斯多夫维数。

Explicit Transition Density Functions of Skew Brownian Motions with Two-Valued Drift

In this article, we derive the explicit transition density functions of skew Brownian motion (SBM in abbreviation) with two-valued drift for all $t>0$. As an important step of this result, it is also shown in this paper that SBM with two-valued drift is a strong Markov process by finding its symmetrizing measure and canonical scale function, from which one can tell what values of the drift make such a process transient or recurrent.

Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

Lenses in Skew Brownian Flow

We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions. Due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, that is, pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. The paper contains qualitative and…

An ideal class to construct solutions for skew Brownian motion equations

This paper contributes to the study of stochastic processes of the class $(Σ)$. First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variational part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class $(Σ)$. More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class $(Σ)$.

On the local time process of a skew Brownian motion

We derive a Ray-Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both these cases. The Ray-Knight type theorem is applied to study integral functionals o…

Resolution of the skew Brownian motion equations with stochastic calculus for signed measures

Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class $Σ(H)$ which appeared for the first time in \cite{f} and studied in \cite{f,e,o}. The second part is devoted to the const…

On countably skewed Brownian motion with accumulation point

In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and po…

A dual skew symmetry for transient reflected Brownian motion in an orthant

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely, that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison, which characterizes the exponential station…

Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition

Motivated by recent developments on random polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. It was shown by Harrison and Williams (1987) that RBM in a polyhe- dral domain has an invariant measure in product form if a certain skew-symmetry condition holds. We show that (modulo technical assumptions) the generalised RBM has an invariant measure in product form if (…

关于 Portenko 对倾斜布朗运动的近似

On Portenko's approximation of skew Brownian motion

本文通过算子半群理论回顾 Portenko 定理，并将其推广到星图上的 Walsh 过程，揭示了倾斜布朗运动边界条件的参数转换机制。

On simultaneous hitting of membranes by two skew Brownian motions

We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.

树配对中的$p$-$θ$关系

The $p$-$θ$ relation in mating of trees and its application to skew Brownian permutons

本文针对树配对方法中耦合两对布朗运动的参数$p$（描述布朗运动）与$θ$（描述LQG曲面上的SLE曲线），推导了二者的精确关系，并将其应用于计算偏置布朗置换子的统计量。

On general skew Brownian motions

The aim of this paper is two-fold. On one hand, we will study the distorted Brownian motion on $\mathbb{R}$, i.e. the diffusion process $X$ associated with a regular and strongly local Dirichlet form obtained by the closure of $\mathscr{E}(f,g)=\frac{1}{2}\int_\mathbb{R} f'(x)g'(x)ρ(x)dx$ for $f,g\in C_c^\infty(\mathbb{R})$ on $L^2(\mathbb{R}, \mathfrak{m})$, where $\mathfrak{m}(dx)=ρ(x)dx$ and $ρ$ is a certain positive function. After figuring out the irreducible decomposition of $X$, we will present a characterization of that $X$ becomes a semi-martingale by virtue of so-called Fukushima's d…

Phase transition for parking blocks, Brownian excursion and coalescence

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n < m), and l = m<n empty places. For a non computer science-minded reader, we shall use the metaphore of n cars parking on m places: each car chooses a place at random, and if this place k is occupied, the car tries successively k+1, k+2, ... until it finds an empty place (with the convention that place m+1 is actually place 1). Pittel [42] proves that when l/m goes to some positive limit a < 1, the size of the largest block of consecutive cars is O(log m). In this paper we examine at which leve…

Skew Brownian diffusions across Koch interfaces

We consider planar skew Brownian motion (BM) across pre-fractal Koch interfaces $\partial Ω^n$ and moving on $\overline{Ω^n} \cup Σ^n= Ω^n_\varepsilon$. We study the asymptotic behaviour of the corresponding multiplicative additive functionals when thickness of $Σ^n$ and skewness coefficients vanish with different rates.

关于折射偏布朗运动及其应用的一个注记

A note on Refracted Skew Brownian Motion with an application

本文采用扰动方法求出折射偏布朗运动的势密度表达式，恢复其转移密度并研究长时间渐近行为，同时提出两种生成准随机样本的方法以应用于风险测量。

Pointwise weak existence of distorted skew Brownian motion with respect to discontinuous Muckenhoupt weights

For any starting point in $\mathbb{R}^d$, we identify the stochastic differential equation that is satisfied by distorted Brownian motion with respect to a certain discontinuous Muckenhoupt $A_2$-weight $ψ$. The discontinuities of $ψ$ typically take place on a sequence of level sets of the Euclidean norm $D_k:=\{x\in \mathbb{R}^d\, | \;\|x\|=d_k\}$, $k\in\mathbb{Z}$, where $(d_k)_{k\in\mathbb{Z}}\subset (0,\infty)$ may have accumulation points and each level set $D_k$ plays the role of a permeable membrane.

德芬尼控制下的折射斜布朗运动

De Finetti's Control for Refracted Skew Brownian Motion

本文提出折射斜布朗运动作为风险模型，研究其最优分红策略的充分条件。

Duality between coalescence times and exit points in last-passage percolation models

In this paper we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy process.

On a skew stable Lévy process

The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in the situation that the noise is a stable Lévy process with finite mean and infinite variance. We define a skew stable Lévy process $X$ as the limit of a sequence of stable Lévy processes which are perturbed at zero. We point out a formula for the resolvent of $X$ and show that $X$ is a solution to a stochastic differential equation with a local time. Also, we…

一维扩散的极限定理：反射、偏斜、黏性、偏斜-黏性及断裂极限

Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits

本文建立正则一维扩散弱收敛的一般条件，基于Itô–McKean表示将随机收敛化为确定性问题，并应用于半透膜介质均匀化，得到反射、偏斜、黏性及断裂等极限过程。

Density of Skew Brownian motion and its functionals with application in finance

We derive the joint density of a Skew Brownian motion, its last visit to the origin, local and occupation times. The result is applied to option pricing in a two valued local volatility model and in a displaced diffusion model with constrained volatility.

强Baxter与半Baxter排列的置换极限是斜布朗置换

The permuton limit of strong-Baxter and semi-Baxter permutations is the skew Brownian permuton

本文证明强Baxter和半Baxter排列在置换意义下收敛于非退化斜布朗置换，并开发了基于生成树和锥中多维游走的通用收敛证明技术。

Corrections for "Occupation and local times for skew Brownian motion with applications to dispersion across an interface"

We are making corrections and acknowledging colleagues that pointed out mistakes in our recent paper titled "Occupation and local times for skew Brownian motion with applications to dispersion across an interface" which was published in Annals of Applied Probability (2011) 21(1) 183-214. Specifically the corrections are: 1. The restriction of $γ$ to non negative values in Theorem 1.3 is not needed. But one has probabilistic interpretation only when $γ$ is non negative. 2. State the correct formulas in Corollary 3.3 as their were computational errors in the original formulas. We thank Pierre Et…

On pairs of complementary transmission conditions and on approximation of skew Brownian motion by snapping-out Brownian motions

Following our previous work on `perpendicular' boundary conditions, we show that transmission conditions \[ f'(0-)=α(f(0+)-f(0-)), \quad f'(0+)=β(f(0+)-f(0-)),\] describing so-called snapping out Brownian motions on the real line, are in a sense complementary to the transmission conditions \[f(0-)=-f(0+), \quad f''(0+) =αf'(0-)+βf'(0+). \] As an application of the analysis leading to this result, we also provide a deeper semigroup-theoretic insight into the theorem saying that as the coefficients $α$ and $β$ tend to infinity but their ratio remains constant, the snapping-out Brownian motions c…

Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of p…

Is a Brownian skew?

We study the asymptotic behavior of the maximum likelihood estimator corresponding to the observation of a trajectory of a Skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non-classical, under the null hypothesis of the Skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations that can be easily performed to estimate the skewness parameter, and provide an application in Bi…

非局部偏斜 Brownian 运动与非局部粘性 Brownian 运动

Non-local skew and non-local sticky Brownian motions

本文通过在实轴热方程原点引入非局部算子，推广了偏斜与双侧粘性 Brownian 运动，提出了非局部偏斜与粘性 Brownian 运动两种随机过程，并揭示了其跳跃与非马尔可夫动力学特性。

A Pluzhnikov's Theorem, Brownian motions and Martingales in Lie Group with skew-symmetric connections

Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem.

从偏斜到反射的SPDE

From skew to reflected SPDEs

本文证明偏斜随机热方程的解在无限偏斜极限下收敛到反射随机热方程的解，建立了偏斜与反射布朗运动联系的SPDE类比。

Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

We investigate a class of optimal stopping problems arising in, for example, studies considering the timing of an irreversible investment when the underlying follows a skew Brownian motion. Our results indicate that the local directional predictability modeled by the presence of a skew point for the underlying has a nontrivial and somewhat surprising impact on the timing incentives of the decision maker. We prove that waiting is always optimal at the skew point for a large class of exercise payoffs. An interesting consequence of this finding, which is in sharp contrast with studies relying on …

Optimal stopping of oscillating Brownian motion

We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $σ_1$ and $σ_2$ denote the volatilities on the negative and positive half-lines, respectively. Our main result is that continuation region of the optimal stopping problem with reward $((1+x)^+)^2$ is disconnected, if and only if $σ_1^2<σ_2^2<2σ_1^2$. Based on the fact that the skew Brownian motion in natural scale is an oscillating Brownian motion, the obtained results are translated into corresponding results for the skew Brownian…

Skew Brownian Motion and Complexity of the ALPS Algorithm

Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density π. The paper [24] introduced the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper, we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that under appropriate assumptions, the ALPS algorithm mixes in time O(d[log(d)]^2 ) or O(d), depending on which version is used.

振荡布朗运动与偏斜布朗运动局部时间的收敛速率

Rates of convergence to the local time of Oscillating and Skew Brownian Motions

本文基于高频观测研究振荡与偏斜布朗运动统计量向局部时间的收敛速率，并将其应用于偏斜参数估计和扩散系数估计，首次给出了偏斜参数估计量的收敛速率。

When do skew-products exist?

The classical skew-product decomposition of planar Brownian motion represents the process in polar coordinates as an autonomously Markovian radial part and an angular part that is an independent Brownian motion on the unit circle time-changed according to the radial part. Theorem 4 of Liao (2009) gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action of a Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian "radial" part that lives on the …

Maximum likelihood estimator for skew Brownian motion: the convergence rate

We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order $1/4$. We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In par…

Skew Brownian motion with dry friction: The Pugachev-Sveshnikov approach

The Caughey-Dieness process, also known as the Brownian motion with two valued drift, is used in theoretical physics as an advanced model of the Brownian particle velocity if the resistant force is assumed to be dry friction. This process also appears in many other fields, such as applied physics, mechanics, astrophysics, and pure mathematics. In the present paper we are concerned with a more general process, skew Brownian motion with dry friction. The probability distribution of the process itself and of its occupation time on the positive half line are studied. The approach based on the Puga…

On Tightness of the Skew Random Walks

The primary purpose of this article is to prove a tightness of skew random walks. The tightness result implies, in particular, that the skew Brownian motion can be constructed as the scaling limit of such random walks. Our proof of tightness is based on a fourth-order moment method.

灾害影响人口波动性建模：一种非齐次几何-偏斜布朗运动方法

Modeling Volatility of Disaster-Affected Populations: A Non-Homogeneous Geometric-Skew Brownian Motion Approach

本文针对自然灾害死亡人数预测问题，提出一种整合历史数据、处理数据偏斜并适应时间依赖性的时变偏斜布朗运动模型，并证明了解的存在唯一性，为灾害风险波动性评估提供了新方法。

A note on the exact simulation of spherical Brownian motion

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.

Optimal stopping of Brownian motion with broken drift

We solve an optimal stopping problem where the underlying diffusion is Brownian motion on $\bf R$ with a positive drift changing at zero. It is assumed that the drift $μ_1$ on the negative side is smaller than the drift $μ_2$ on the positive side. The main observation is that if $μ_2-μ_1>1/2$ then there exists values of the discounting parameter for which it is not optimal to stop in the vicinity of zero where the drift changes. However, when the discounting gets bigger the stopping region becomes connected and contains zero. This is in contrast with results concerning optimal stopping of skew…

凝聚对称包含过程在环面上热力学极限下的收敛性到凝聚布朗运动

Convergence of the Condensing Symmetric Inclusion Process on the Torus in the Thermodynamical Limit to Coalescing Brownian Motions

研究了离散一维环面上凝聚对称包含过程的饱和态，证明在适当缩放下，凝聚体位置收敛于连续环面上的凝聚布朗运动，通过控制凝聚时间并结合无凝聚运动的精确估计，证明其收敛性。

斜布朗置换子：随机约束置换的一个新普适类

The skew Brownian permuton: a new universality class for random constrained permutations

本文构造了由两个参数刻画的斜布朗置换子族，统一了Baxter置换子与有偏布朗可分置换子，并证明其与SLE装饰的LQG球面相关置换子几乎覆盖全部参数范围。

Stochastic flows related to Walsh Brownian motion

We define an equation on a simple graph which is an extension of Tanaka equation and the skew Brownian motion equation. We then apply the theory of transition kernels developped by Le Jan and Raimond and show that all the solutions can be classified by probability measures.

在支撑位买入：多偏斜布朗运动下的入场问题

Buying at Support: The Entry Problem under Multi-Skew Brownian Motion

本文研究在几何多偏斜布朗运动价格模型下，考虑后续最优卖出的买入时机问题，证明最优买入仅在价格被向上推动的水平发生，并给出单支撑单阻力情形下在支撑位买入的显式最优规则。

多偏斜布朗运动下带支撑位与阻力位的最优清算

Optimal Liquidation with Support and Resistance Levels under Multi-Skew Brownian Motion

本文研究多偏斜布朗运动下带支撑与阻力位的最优清算问题，提出三种几何形态的闭式准则，并揭示向阻力位卖出为结论及参数识别的不对称性。

A scaling proof for Walsh's Brownian motion extended arc-sine law

We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.

直线及两个闭合半轴上最一般的布朗运动

The Most General Brownian Motion on the Line and on Two Closed Half-Lines

本文研究了直线及两个闭合半轴上最一般的布朗运动，提出并证明了零点杀死斜粘性布朗运动及其扩展形式。

On stochastic continuity of generalized diffusion processes constructed as the strong solution to an SDE

The comparison theorem for skew Brownian motions is proved. As the corollary we get the estimate on ${\Cal L}_1-$distance between two skew Brownian motions started from different points. Using this result we prove the continuous dependence on starting point of one class of generalized diffusion processes constructed as the strong solution to an SDE.