A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $Ω$, it is only known that the radius of convergence for the expected signature on sufficiently regular $Ω$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $Ω$ is the $2$-dimensional unit disc ([1]).
In this paper, we prove that on any bounded $C^{2,α}$-domain $Ω\subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.
The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.
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 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.
Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk
University of Zagreb Faculty of Science（萨格勒布大学理学院）；Department of Mathematics and Statistics, Saint Louis University（圣路易斯大学数学与统计系）；University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）
We study a correlated Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the comple…
Iterated Bessel processes R^γ(t), t>0, γ>0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions B^{hp}(t), t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I_R(t)=_1R^γ(_2R^γ(t)), t>0 and J_R(t) =_1R^γ(|_2R^γ(t)|^2), t>0 are obtained and discussed. Processes of the form R^γ(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are examined and numerous probability laws derived, including the Student law, the arcsin laws (also their asymmetric versions), the Lamperti distribution of the r…
We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.
We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.
We discuss the class of "Quadratic Normal Volatility" models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility. We characterize these models as the ones that can be obtained from stopped Brownian motion by a simple transformation and a change of measure that only depends on the terminal value of the stopped Brownian motion. This explains the existence of explicit analytic formulas for option prices within Quadratic Normal Volatility models in the academic literature.
The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient betwee…
Brownian motion in the plane in the presence of a "trap" at which motion is stopped is studied. If the trap $T$ is a connected compact set, it is shown that the probability for planar Brownian motion to hit this set before a given time $t$ is well approximated even at short times by the probability that Brownian motion hits a disk of radius $r_T$ equal to the conformal radius of the trap $T$.
For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $Δ$ and stopping it at discrete times $(iΔ)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*σ|(t,x)\sqrt Δ$ where $σ$ stands for the diffusion coefficient of the process. The procedure is thus extremel…
Let $τ$ be the first hitting time of the point 1 by the geometric Brownian motion $X(t)= x \exp(B(t)-2μt)$ with drift $μ\geq 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from 0 with $E^0 B^2(t) = 2t$. We provide an integral formula for the density function of the stopped exponential functional $A(τ)=\int_0^τX^2(t) dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in \cite{BGS}, the present paper also covers the case of arbitrary drifts $μ\geq 0$ and provides a significant unification and extension of results of the ab…
The signature of a path provides a top down description of the path in terms of its effects as a control [Differential Equations Driven by Rough Paths (2007) Springer]. The signature transforms a path into a group-like element in the tensor algebra and is an essential object in rough path theory. The expected signature of a stochastic process plays a similar role to that played by the characteristic function of a random variable. In [Chevyrev (2013)], it is proved that under certain boundedness conditions, the expected value of a random signature already determines the law of this random signa…
We find a lower bound for the Hausdorff dimension that a Liouville Brownian motion spends in $α$-thick points of the Gaussian Free Field, where $α$ is not necessarily equal to the parameter used in the construction of the geometry. This completes a conjecture in \cite{berestycki2013diffusion}, where the corresponding upper bound was shown. In the course of the proof, we obtain estimates on the (Euclidean) diffusivity exponent, which depends strongly on the nature of the starting point. For a Liouville typical point, it is $1/(2 - \frac{γ^2}{2})$. In particular, for $γ> \sqrt{2}$, the path is L…
In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in bounded kappa-fat open set; if u is a positive harmonic function with respect to X in a bounded kappa-fat open set D and h is a positive harmonic function in D vanishing on D^c, then the non-tangential limit of u/h exists almost …
We prove Fatou's theorem for nonnegative harmonic functions with respect to subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential conver…
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models a…
In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire \…
We consider one-dimensional excited random walks (ERWs) with i.i.d. markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie markov chain at a single site. While our results extend the results of Dolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) and Kosygina and Peterson (2016, ERWs with periodic cookie stacks), the a…
We present a constructive probabilistic proof of the fact that if $B=(B_t)_{t\ge0}$ is standard Brownian motion started at $0$, and $μ$ is a given probability measure on $\mathbb{R}$ such that $μ(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $τ_{b,c}=\inf\{t>0\vert B_t\ge b(t)$ or $B_t\le c(t)\}$ has the law $μ$. The method of proof relies upon weak convergence arguments arising from Helly's s…
Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process
Graduate School of Science, Kyoto University（京都大学大学院理学研究科）；Department of Business Economics, School of Management, Tokyo University of Science（东京理科大学经营学部商业经济学科）
On the cover time of Brownian motion on the Brownian continuum random tree
Department of Computer Science, New York University Abu Dhabi（阿布扎比纽约大学计算机系）；Research Institute for Mathematical Sciences, Kyoto University, Japan（日本京都大学数学科学研究所）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）；Modal’X, UMR CNRS 9023, UPL, Univ. Paris-Nanterre, F92000 Nanterre, France（法国楠泰尔巴黎西奈大学Modal’X实验室）
We consider an obliquely reflected Brownian motion $Z$ with positive drift in a quadrant stopped at time $T$, where $T:=\inf \{ t>0 : Z(t)=(0,0) \}$ is the first hitting time of the origin. Such a process can be defined even in the non-standard case where the reflection matrix is not completely-$\mathcal{S}$. We show that in this case the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in a finite time. Given an arbitrary starting point $(u,v)$ in the quadrant, we consider the escape (resp. absorption) probabilities $\mathbb{P}_{(u,v)}[T=\infty]$ …
It is well-known (see Dvoretzky, Erd{\H o}s and Kakutani [8] and Le Gall [12]) that a planar Brownian motion $(B_t)_{t\ge 0}$ has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the construction of a family of random measures, denoted by $\{{\mathcal M}_{\infty}^α\}_{0< α<2}$, that are supported by the set of the points of infinite multiplicity. We prove that for any $α\in (0, 2)$, almost surely the Hausdorff dimension of ${\mathcal M}_{\infty}^α$ equals $2-α$, and ${\mathcal M}_{\infty}^α$ is supported by the set of thick points defined in B…
In a recent work by Gross, the following problem was stated and solved: given a measure $μ$ with finite second moment, find a simply connected domain $U$ in $\CC$ such that the real part of a Brownian motion stopped when it leaves $U$ is distributed as $μ$. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this…
