We present a numerical framework for approximating the $μ$-domain in the planar Skorokhod embedding problem PSEP, recently introduced in \cite{gross2019}. We show that under weak convergence of a sequence of probability measures $(μ_{n})_{n}$, the corresponding sequence of $μ_{n}$-domains converges, in an appropriate sense, to the domain associated with the limit measure $μ$. In addition, we provide implementation strategies, convergence rate estimates, and a numerical example. The method is robust and versatile, offering a concrete computational approach for the approximation of $μ$-domains. As part of this analysis, we introduce a novel mode of convergence for planar domains via planar Brownian motion, which we call $p$-Brownian convergence.
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.
We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.
The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple symmetric random walk and when no external randomisation is allowed. We prove that any measure on Z can be embedded by means of a minimal stopping time. However, in sharp contrast to the Brownian setting, we show that the set of measures which can be embedded in a uniformly integrable way is strictly smaller then the set of centered probability measures: specifically it is a fractal set which we characterise as an iterated function system…
Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the x coordinate at the hitting time has any prescribed centered distribution with finite variance. This provides a new, complex-analytic proof of the Skorokhod embedding theorem. Our method is constructive and can give an explicit description of the barrier.
In the recent papers [Lochowski:2011fk, Lochowski:2013yq, Lochowski:2013lr] the truncated variation has been introduced, characterized and studied in various stochastic settings. In this note we uncover an intimate link to the Skorokhod problem. Further, we exploit it to give an explicit representation of the truncated variation of a Brownian motion. More precisely, we prove that the inverse of this process is, up to a minor time shift, a Lévy subordinator with the exponent \sqrt{2q}\tanh(c\sqrt{q/2}) . This also gives a representation of a solution of the two-sided Skorokhod problem for a Bro…
We give a simple proof that in a Lipschitz domain in two dimensions with Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation governing reflecting Brownian motion.
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.
We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.
In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target l…
Given a Brownian motion $B_t$ and a general target law $μ$ (not necessarily centered or even integrable) we show how to construct an embedding of $μ$ in $B$. This embedding is an extension of an embedding due to Perkins, and is optimal in the sense that it simultaneously minimises the distribution of the maximum and maximises the distribution of the minimum among all embeddings of $μ$. The embedding is then applied to regular diffusions, and used to characterise the target laws for which a $H^p$-embedding may be found.
The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optim…
Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq \mathbb R$ and let $μ$ be a probability measure on $I^X$. Then $τ$ is a solution of the Skorokhod embedding problem (SEP) for $μ$ in $X$ if $τ$ is a stopping time and $X_τ\sim μ$. There are well-known conditions which determine whether there exists a solution of the SEP for $μ$ in $X$. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution of the SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first m…
We provide a new probabilistic proof of the connection between Rost's solution of the Skorokhod embedding problem and a suitable family of optimal stopping problems for Brownian motion with finite time-horizon. In particular we use stochastic calculus to show that the time reversal of the optimal stopping sets for such problems forms the so-called Rost's reversed barrier.
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.
Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an non-negative orthant, specifically in the context of queueing networks in the heavy traffic regime. One of the key problems is identifying conditions for stability of a Skorokhod problem, defined as the property that trajectories are attracted to the origin. The stability conditions are known in dimension up to three, but not for general dimensions. In this paper we explain the fundamental difficulties encountered in trying to establish stability conditions for general dimensions. We prove that stability of Skorokhod pr…
The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset\mathbb{C}$ whose exit time embeds a given probability distribution $μ$ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $μ$ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb{U}$ …
Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod's selection theorem.
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod embedding problem to the case of finitely-many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results which are formulated by means of probability measures on an enlarged space. We also relate these results to the problem of martingale optimal transport under multiple marginal cons…
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.
The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging because the dynamics are discontinuous at the boundary of the domain and the boundary of the domain is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected version. In this work we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz …
Motivated by the model- independent pricing of derivatives calibrated to the real market, we consider an optimization problem similar to the optimal Skorokhod embedding problem, where the embedded Brownian motion needs only to reproduce a finite number of prices of Vanilla options. We derive in this paper the corresponding dualities and the geometric characterization of optimizers. Then we show a stability result, i.e. when more and more Vanilla options are given, the optimization problem converges to an optimal Skorokhod embedding problem, which constitutes the basis of the numerical computat…
In this paper, we provide some results on Skorokhod embedding with local time and its applications to the robust hedging problem in finance. First we investigate the robust hedging of options depending on the local time by using the recently introduced stochastic control approach, in order to identify the optimal hedging strategies, as well as the market models that realize the extremal no-arbitrage prices. As a by-product, the optimality of Vallois' Skorokhod embeddings is recovered. In addition, under appropriate conditions, we derive a new solution to the two-marginal Skorokhod embedding as…
We study the problem of stopping a Brownian motion at a given distribution $ν$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(ν)$ of stopping times embedding $ν$ is weakly dense in the set $\mathcal{R}(ν)$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $\mathcal{T}(ν)$ has the same value as the relaxed one over $\mathcal{R}(ν)$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kanto…
The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $τ$ so that $W_τ$ is distributed according to a prescribed probability distribution $μ$. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let $X$ be a Markov process with initial marginal distribution $μ_0$ and let $μ_1$ be a probability measure. The task is to find a stopping time $τ$ such that $X_τ$ is distributed according to $μ_1$. More precisely, we study the questio…
We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using Transportation Cost Inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions, and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over lar…
With the help of the Gauss-Laplace transform for the exit time from a cone of planar Brownian motion, we obtain some infinite divisibility properties for the reciprocal of this exit time.
Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 πN 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly, as $N$ tends to infinity, towards the occupation measure of the Brownian motion.
A number of recent new proofs of Euler's celebrated identity are presented, all of which are probabilistic in nature and depend on the conformal invariance of planar Brownian motion.
We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $ξ(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-ξ(A)$. A consequence of this and continuity of $ξ(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.
For every integer $n\geq 1$, we consider a random planar map $\mathcal{M}_n$ which is uniformly distributed over the class of all rooted bipartite planar maps with $n$ edges. We prove that the vertex set of $\mathcal{M}_n$ equipped with the graph distance rescaled by the factor $(2n)^{-1/4}$ converges in distribution, in the Gromov-Hausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map.
In this paper, we construct a counterexample to a question by Cantelli, asking whether there exists a nonconstant positive measurable function $φ$ such that for i.i.d. r.v. $X,Y$ of law $\mathcal{N}(0,1)$, the r.v. $X+φ(X)\cdot Y$ is also Gaussian. This construction is made by finding an unusual solution to the Skorokhod embedding problem (showing that the corresponding Brownian transport, contrary to the Root barrier, is not unique). To find it, we establish some sufficient conditions for the continuity of the Root barrier function.
Known results show that the diameter $d_1$ of the trace of planar Brownian motion run for unit time satisfies $1.595 \leq \mathbb{E} d_1 \leq 2.507$. This note improves these bounds to $1.601 \leq \mathbb{E} d_1 \leq 2.355$. Simulations suggest that $\mathbb{E} d_1 \approx 1.99$.
Prompted by an example arising in critical percolation, we study some reflected Brownian motions in symmetric planar domains and show that they are intertwined with one-dimensional diffusions. In the case of a wedge, the reflected Brownian motion is intertwined with the 3-dimensional Bessel process. This implies some simple hitting distributions and sheds some light on the formula proposed by Watts for double-crossing probabilities in critical percolation.
We obtain a formula for the density of the winding number of planar Brownian motion around the origin, and deduce from it asymptotic expansions in inverse powers of the logarithm of the squared time, explicit in the angular variable. In particular, we obtain the corrections of any order to the Spitzer's law, and also to a local limit theorem for the windings.
In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg Lawler and Oded Schramm) on critical exponents for two-dimensional simple random walks, Brownian motions and other conformally invariant random objects.
The purpose of the paper is to find the joint distribution of the hitting time and place of two-dimensional Brownian motion hitting the negative horizontal axis. We provide various formulas for Green functions as well as for the conditional distributions. In the end we treat an analogous problem for the case when vertical component has a negative drift.
The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane $\mathbb{H}$, obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski.
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary dist…
Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $ϕ_α(s)=log(1/s)^{-α}$ for $α>0$. If $α<1$ we show that almost surely there exists a point x in the plane such that $H^{ϕ_α}({t > 0 : B_t=x})>0$, but if $α>1$ almost surely $H^{ϕ_α} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986.
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author's knowledge, have not been published anywhere. The main focus is on the two-dimensional processes.
It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce s…
In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $ρ$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $ρ$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
