Jean-François Marckert
A theorem of Donsker asserts that the empirical process converges in distribution to the Brownian bridge. The aim of this paper is to provide a new and simple proof of this fact.
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Let $T>0,α>\frac12$. In the present paper we consider the $α$-Brownian bridge defined as $dX_t=-α\frac{X_t}{T-t}dt+dW_t,~ 0\leq t< T$, where $W$ is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $ α$ based on the continuous observation $\{X_s,0\leq s\leq t\}$ as $t\uparrow T$. We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $\frac{1}{\sqrt{|\log(T-t)|}}$, as $t\uparrow T$. First we compute an upper bound and then find a lower bound with the …
We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge together with various relationships between the laws of these four processes.
We study the problem of stopping an $α$-Brownian bridge as close as possible to its global maximum. This extends earlier results found for the Brownian bridge (the case $α=1$). The exact behavior for $α$ close to $0$ is investigated.
We discuss the distributions of three functionals of the free Brownian bridge: its $Ł^2$-norm, the second component of its signature and its Lévy area. All of these are freely infinitely divisible. We introduce two representations of the free Brownian bridge as series of free semicircular random variables are used, analogous to the Fourier representations of the classical Brownian bridge due to \ts{Lévy} and \ts{Kac}.
In a recent work J. Pitman and W. Tang defined the Vervaat's transform for a Brownian bridge with two different endpoints and for a Brownian motion between times $0$ and $1$. They proved some path decomposition properties for these Vervaat's transforms and raised open questions on their semi-martingale decomposition. In this paper we give an alternative proof for the path decomposition and answer the open questions.
We establish the Brownian bridge asymptotics for a scaled self-avoiding walk conditioned on arriving to a far away point $n \vec{a}$ for $\vec{a}$ in $Z^d$, as $n$ increases to infinity.
Convex hull of Brownian motion and Brownian bridge
本文研究平面布朗运动与独立布朗桥轨迹并集的凸包，给出其周长与面积期望的精确值及联合最大值时刻密度，并推广到多个独立过程。
In this paper we study periodical stochastic processes, and we define the conditions that are needed by a model to be a good noise model on the circumference. The classes of processes that fit the required conditions are studied together with their expansion in random Fourier series in order to provide results about their path regularity. Finally, we discuss a simple and flexible parametric model with prescribed regularity that is used in applications, and we prove the asymptotic properties of the maximum likelihood estimates of model parameters.
We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.
We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole.
We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three probability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $Π(G)$ of the gap between the maxima, $G = M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson …
The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered.
Rate of convergence of the conditioned random walk towards the Brownian bridge
本文通过结合Stein方法和Radon-Nikodym表示，研究了条件随机游走和经验过程向布朗桥收敛的速度并给出了距离界。
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.
A multi-player optimal stopping game based on a spider Brownian bridge
研究由蜘蛛布朗桥驱动的多人非零和最优停止博弈，先分析两人情形下纳什均衡存在及对称情形下唯一性的条件并举例说明多均衡情况，后扩展到多人情形并给出相应存在条件。
We observe that the probability distribution of the Brownian motion with drift $-c \frac x {1-t}$ where $c\not =1$ is singular with respect to that of the classical Brownian bridge measure on $[0,1]$, while their Cameron-Martin spaces are equal set-wise if and only if $c> \frac 12$, providing also examples of exponential martingales on $[0,1)$ not extendable to a continuous martingale on $[0,1]$. Other examples of generalised Brownian bridges are also studied.
We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1}, u\leq 1)$ under uniform random sampling.
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate.
Pitman's and Lévy's theorems for Brownian bridges
研究具有任意端点的一维布朗桥的皮特曼和列维定理，采用综合研究方法。
On the excision of Brownian bridge paths
本文通过切除布朗桥路径的特定游走，建立了与三维贝塞尔桥的联系，拓展了布朗运动与贝塞尔过程之间的理论联系。
We study the law of the minimum of a Brownian bridge, conditioned to take specific values at specific points, and the law of the location of the minimum. They are used to compare some non-adaptive optimisation algorithms for black-box functions for which the Brownian bridge is an appropriate probabilistic model and only a few points can be sampled.
Absorption Times for Discrete Whittaker Processes and Non-Intersecting Brownian Bridges
研究离散惠特克过程吸收时间与非相交布朗桥最大高度关系，给出二者推测关系的证据。
For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(τ(f)+t \mod1)+f(1)1_{\{t+τ(f) \geq 1\}}-f(τ(f))$, where $τ(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms for random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decompositions and study its semimartingale properties. …
Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the r…
The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joini…
Online minimum search for a Brownian bridge
本文研究数值求解Brownian bridge最小值和arg-min的计算问题，基于Pitman等人的著名结果，证明了二分法具有较小的误差和失败概率。
For the d-dimensional model of a subcritical bond percolation (p<p_c) and a point \vec{a} in Z^d, we prove that a cluster conditioned on connecting points (0,...,0) and n\vec{a} if scaled by 1/(n|vec{a}|) along \vec{a} and by 1/sqrt{n} in the orthogonal direction converges asymptotically to Time x (d-1)-dimensional Brownian Bridge.
We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold.
Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.
This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.
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 $t$.
For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(τ(f)+t \mod1)+f(1)1_{\{t+τ(f) \geq 1\}}-f(τ(f))$, where $τ(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale proper…
Optimal Stopping of a Brownian Excursion and an $α$-dimensional Bessel Bridge
研究$α$维Bessel桥在收益$ϕ(x)=x^n$下的最优停止问题，对布朗游走特例给出显式解，一般情形则通过幂级数展开提供完整解。
I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.
Local Statistics and Concentration for Non-intersecting Brownian Bridges With Smooth Boundary Data
本文研究一般边界条件下非相交布朗桥的集中性与局部统计，在光滑极限形状假设下证明了近乎最优集中界，并证明其体局部统计沿任意固定时间收敛到 sine 过程。
Construction and sample path properties of Brownian house-moving between two curves
本研究构造了被限制在两条曲线之间的“布朗搬家”随机过程，通过带条件布朗运动、布朗桥及三维贝塞尔桥的弱收敛相关命题实现构造，并研究了其样本轨道性质，丰富了带约束随机过程的相关研究。
We introduce an infinite time horizon Brownian bridge which is determined by a stochastic Langevin equation with time dependent drift coefficient. We show that this process goes to zero almost surely when the time goes to infinity and study the existence and asymptotic behavior of its local time as well as its Hölder continuity in time variable and in location variable. The main difficulty is the lack of stationarity of the process so that the powerful tools for stationary (Gaussian) processes are not applicable. We employ the Garsia-Rodemich-Rumsey inequality to get around this type of diffic…
This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of the these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties.
In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation o…
Self-repellent Brownian Bridges in an Interacting Bose Gas
该研究提出d维相互作用量子玻色气体的布朗桥系综模型，推导热力学极限下的自由能公式与凝聚相变判据，证明d≥5且弱相互作用时凝聚相非空，方法结合自排斥随机游走特性与蕾丝展开技术。
For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.
We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle, and one of us. Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time.
We present a simulation scheme for simulating Brownian bridges on complete and connected Lie groups. We show how this simulation scheme leads to absolute continuity of the Brownian bridge measure with respect to the guided process measure. This result generalizes the Euclidean result of Delyon and Hu to Lie groups. We present numerical results of the guided process in the Lie group $\SO(3)$. In particular, we apply importance sampling to estimate the metric on $\SO(3)$ using an iterative maximum likelihood method.
The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosén and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.
The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.
Let v be a bounded function with bounded support in R^d, d>=3. Let x,y in R^d. Let Z(t) denote the path integral of v along the path of a Brownian bridge in R^d which runs for time t, starting at x and ending at y. As t->infty, it is perhaps evident that the distribution of Z(t) converges weakly to that of the sum of the integrals of v along the paths of two independent Brownian motions, starting at x and y and running forever.Here we prove a stronger result, namely convergence of the corresponding moment generating functions and of moments. This result is needed for applications in physics.
We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on $[0,1]$ pinned down in 0 at time 1 and cond…
In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le τ\le 1}\mathsf{E}[\mathrm{e}^{X_τ}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we devel…
Let $(B(t))_{t\in [0,1]}$ be the linear Brownian motion and $(X_n(t))_{t\in [0,1]}$ be the $(n-1)$-fold integral of Brownian motion, $n$ being a positive integer: $$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$ In this paper we construct several bridges between times 0 and 1 of the process $(X_n(t))_{t\in [0,1]}$ involving conditions on the successive derivatives of $X_n$ at times 0 and 1. For this family of bridges, we make a correspondance with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical pr…
We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schrödinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in [20], and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the measure constructed in [24].
Parameter Estimation for Complex α-Fractional Brownian Bridge
本文研究复α分数布朗桥过程的参数估计问题，通过建立良好定义性和证明估计量的强一致性和渐近分布，揭示了二维极限分布的非柯西边缘特性。
We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(BB_t)_{0 \leq t \leq 1}$ from $0$ to $0$ in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker's delta (which correlates with the local time of $|BB|$ at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.
The issue of giving an explicit description of the flow of information concerning the time of bankruptcy of a company (or a state) arriving on the market is tackled by defining a bridge process starting from zero and conditioned to be equal to zero when the default occurs. This enables to catch some empirical facts on the behavior of financial markets: when the bridge process is away from zero, investors can be relatively sure that the default will not happen immediately. However, when the information process is close to zero, market agents should be aware of the risk of an imminent default. I…
We consider non-colliding Brownian bridges starting from two points and returning to the same position. These positions are chosen such that, in the limit of large number of bridges, the two families of bridges just touch each other forming a tacnode. We obtain the limiting process at the tacnode, the "asymmetric tacnode process". It is a determinantal point process with correlation kernel given by two parameters: (1) the curvature's ratio λ>0 of the limit shapes of the two families of bridges, (2) a parameter σcontrolling the interaction on the fluctuation scale. This generalizes the result f…
Brownian Motion with a Pulse: A Biostatistician's Guide to Diffusions, Bridges, Functional PCA, and First-Passage Models
本文教程系统介绍布朗运动在生物统计中的应用，涵盖构造、路径性质、马尔可夫性、Karhunen-Loeve展开、函数主成分分析、反射原理、局部时、随机微分方程、布朗桥及经验过程极限，并通过纵向生物标志物、退化建模等实例展示其应用。
We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde μ({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study …
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $μ_{A_n}=(1/n)\sum_{i}δ_{a_i}, μ_{B_n}=(1/n)\sum_i δ_{b_i}$ are discretization of probability measures $μ_A, μ_B$. Under regularity assumptions of $μ_A, μ_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
Let alpha,T>0. We study the asymptotic properties of a least squares estimator for the parameter alpha of a fractional bridge defined as dX_t=-alpha*X_t/(T-t)dt+dB_t, with t in [0,T) and where B is a fractional Brownian motion of Hurst index H>1/2. Depending on the value of alpha, we prove that we may have strong consistency or not as t tends to T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W.
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>…
The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the invol…
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