Penalizations of Walsh Brownian motion

Penalizations of the Brownian motion by a functional of its local times

In this article, we study the family of probability measures (indexed by a positive real number t), obtained by penalization of the Brownian motion by a given functional of its local times at time t. We prove that this family tends to a limit measure when t goes to infinity if the functional satisfies some conditions of domination, and we check these conditions in several particular cases.

瓦尔什布朗运动与受扰随机游走的邓斯克缩放极限

Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks

本文研究了受扰随机游走的邓斯克缩放极限，证明其收敛于具有特定权重的瓦尔什布朗运动，并展示了其在膜上的振荡斜布朗运动特性。

Scaled penalization of Brownian motion with drift and the Brownian ascent

We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor in arXiv:math/0511102. The original model penalizes Brownian motion with drift $h\in\mathbb{R}$ by the weight process ${\big(\exp(νS_t):t\geq 0\big)}$ where $ν\in\mathbb{R}$ and $\big(S_t:t\geq 0\big)$ is the running maximum of the Brownian motion. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the $(ν,h)$-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scalin…

作为时间变换Walsh布朗运动的星图上的一般扩散过程

General diffusions on the star graph as time-changed Walsh Brownian motion

本研究将星图上的一般正则扩散表示为时间变换Walsh布朗运动，据此推导粘性粘合扩散的时间变换表示、占据时间公式，还拓展了Feller相关判据并证明星图Dirichlet问题解的存在唯一性及显式形式。

Embedding of Walsh Brownian Motion

Let $(Z,κ)$ be a Walsh Brownian motion with spinning measure $κ$. Suppose $μ$ is a probability measure on $\mathbb{R}^n$. We characterize all the $κ$ such that $μ$ is a stopping distribution of $(Z,κ)$. If we further restrict the solution to be integrable, we show that there would be only one choice of $κ$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[Ψ(L^Z_τ)]$ among all the admissible solutions $τ$, where $Ψ$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.

Serve the shortest queue and Walsh Brownian motion

We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^N$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the …

Stochastic Integral Equations for Walsh Semimartingales

We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison-Shepp-type equations and a change-of-variable formula in the spirit of Freidlin-Sheu for these so-called "Walsh semimartingales". We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.

Limiting laws for long Brownian Bridges perturbed by their one-sided maximum, III

Results of penalization of a one-dimensional Brownian motion $(X_t) $, by its one-sided maximum $\dis (S_t=\sup_{0 \leq u \leq t}X_u)$, which were recently obtained by the authors are improved with the consideration-in the present paper- of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\to \infty$, or penalizations by functions of $(S_t,X_t)$, and also the study of the speed of convergence, as $t\to \infty$, of the penalized distributions at time $t$.

Some limiting laws associated with the integrated Brownian motion

We study some limit theorems for the normalized law of integrated Brownian motion perturbed by several examples of functionals: the first passage time, the nth passage time, the last passage time up to a finite horizon and the supremum. We show that the penalization principle holds in all these cases and give descriptions of the conditioned processes. In particular, it is remarkable that the penalization by the nth passage time is independent of n, and always gives the same conditioned process, i.e. integrated Brownian motion conditioned not to hit 0. Our results rely on some explicit formulae…

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 the diameter of the stopped spider process

We consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_τ\leq C_n \sqrt{E τ},$$ where $τ$ is the class of all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.

Reflected BSDE driven by G-Brownian motion with an upper obstacle

In this paper, we study the reflected backward stochastic differential equation driven by G-Brownian motion (reflected G-BSDE for short) with an upper obstacle. The existence is proved by approximation via penalization. By using a variant comparison theorem, we show that the solution we constructed is the largest one.

Brownian Motions on Metric Graphs with Non-Local Boundary Conditions I: Characterization

A classification for Brownian motions on metric graphs, that is, right continuous strong Markov processes which behave like a one-dimensional Brownian motion on the edges and feature effects like Walsh skewness, stickiness and jumps at the vertices, is obtained. The Feller property of these processes is proved, and the boundary conditions of their generators are identified as non-local Feller-Wentzell boundary conditions. By using a technique of successive revivals, a complete description of the generator is achieved for Brownian motions on star graphs.

Reflected Solutions of BSDEs Driven by G-Brownian Motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections

In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.

A unifying class of Skorokhod embeddings: connecting the Azema-Yor and Vallois embeddings

In this paper we consider the Skorokhod embedding problem in Brownian motion. In particular, we give a solution based on the local time at zero of a variably skewed Brownian motion related to the underlying Brownian motion. Special cases of the construction include the Azema-Yor and Vallois embeddings. In turn, the construction has an interpretation in the Chacon-Walsh framework.

Reflected BSDEs driven by G-Brownian motion with non-Lipschitz coefficients

In this paper, we consider the reflected backward stochastic differential equations driven by G-Brownian motion (reflected G-BSDEs) whose coefficients satisfy the beta-order Mao's condition. The uniqueness is obtained by some a priori estimates and the existence can be proved by two different methods. The first one is Picard iteration and the second one is approximation via penalization. The latter construction is useful to get the comparison theorem.

Reflected stochastic differential equations driven by $G$-Brownian motion in non-convex domains

In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $α$-Hölder continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument.

On first exit times and their means for Brownian bridges

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

On a coupling of solutions to the interface SDE on a star graph

Inspired by Tsirelson proof of the non Brownian character of WalshBrownian motion ltration on three or more rays, we prove some results on aparticular coupling of solutions to the interface SDE on a star graph, recentlyintroduced. This coupling consists in two solutions which are independentgiven the driving Brownian motion. As a consequence, we deduce that if the stargraph contains 3 or more rays, the argument of the solution at a xed time isindependent of the driving Brownian motion.

A conservative evolution of the Brownian excursion

We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0,1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution.

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumit…

Brownian Motions on Star Graphs with Non-Local Boundary Conditions

Brownian motions on star graphs in the sense of Itô-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for …

由 $G$-Brownian 运动驱动且具有对角生成元的多维反射倒向随机微分方程

Multi-dimensional reflected BSDEs driven by $G$-Brownian motion with diagonal generators

本文研究由 $G$-Brownian 运动驱动且生成元为对角型的多维反射 $G$-BSDEs 的适定性，利用惩罚法与 Picard 迭代证明解的存在唯一性，并揭示其与完全非线性 PDE 系统障碍问题的联系。

具有从自身局部时中选取旋转测度的Walsh蜘蛛过程鞅问题

Martingale problem for a Walsh spider process with spinning measure selected from its own local time

本文研究旋转测度和系数依赖于交汇点局部时的Walsh蜘蛛扩散过程，通过证明鞅问题适定、利用拼接程序构造解并结合局部时Kirchhoff传输条件证明唯一性，还给出了标准布朗运动分支下的显式分布律及广义SDE解的存在唯一性。

Limit behaviour of random walks on $\mathbb Z^m$ with two-sided membrane

We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ are periodic and also depend on the incoming direction to $H$, what makes the membrane $H$ two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a $m$-dimensional diffusion whose first coordinate is a skew Brownian motion and the other $m-1$ coordinates is a Brownian motion with a singular drift controlled b…