The greatest convex minorant of Brownian motion, meander, and bridge

@article{Pitman2010TheGC,
  title={The greatest convex minorant of Brownian motion, meander, and bridge},
  author={Jim Pitman and Nathan Ross},
  journal={Probability Theory and Related Fields},
  year={2010},
  volume={153},
  pages={771-807}
}
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 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… 
View on Springer
arxiv.org
14 Citations
Background Citations
6
Methods Citations
3

14 Citations

Convex minorants of random walks and Lévy processes
This article provides an overview of recent work on descriptions and properties of the Convex minorants of random walks and Levy processes, which summarize and extend the literature on these…
The convex minorant of a Lévy process
We offer a unified approach to the theory of convex minorants of Levy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Levy…
Markovian structure in the concave majorant of Brownian motion
. The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a…
Convex hulls of random walks and their scaling limits
WHEN IS THE CONVEX HULL OF A LÉVY PATH SMOOTH?
We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is…
The Vervaat transform of Brownian bridges and Brownian motion
For a continuous function f 2 C([0;1]), define the Vervaat transform V (f)(t) := f( (f) +t mod 1) +f(1)1ft+ (f) 1g f( (f)), where (f) corresponds to the first time at which the minimum of f is…
When is the convex hull of a L\'evy path smooth?
. We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is…
Topics in Stochastic Geometry Convex Hullls of Random Walks
This course covers a range of results on the convex hull of random walks in the plane and in higher dimension: expected perimeter length in the planar case, expected number of faces on the boundary,…
Convex Hulls of Planar Random Walks
For the perimeter length Ln and the area An of the convex hull of the first n steps of a planar random walk, this thesis study n→∞ mean and variance asymptotics and establish distributional limits.…
$\varepsilon $-strong simulation of the convex minorants of stable processes and meanders
Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of L\'evy processes, which includes subordinated stable and symmetric L\'evy…
...
...

References

SHOWING 1-10 OF 30 REFERENCES
Remarks on the Convex Minorant of Brownian Motion
Recently Groeneboom [1] studied the concave majorant process of a Brownian motion (Bt, t ≤ O). The purpose of this note is to take a fresh look at some of Groeneboom’s results in the context of path…
The Concave Majorant of Brownian Motion
Let St be a version of the slope at time t of the concave majorant of Brownian motion on [0, oo). It is shown that the process S = (1St: t > 0) is the inverse of a pure jump process with independent…
The convex minorant of a Lévy process
We offer a unified approach to the theory of convex minorants of Levy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Levy…
Brownian motion and diffusion
Abstract : The book is part of a trilogy covering the field of Markov processes and provides a readable and constructive treatment of Brownian motion and diffusion. It contains some of the author's…
The distribution of the maximal difference between Brownian bridge and its concave majorant
We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and…
Characterization of the least concave majorant of brownian motion, conditional on a vertex point, with application to construction
The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser,…
Convex minorants of random walks and Brownian motion
Let $(S_{i})_{i=0}^n$ be the random walk process generated by a sequence of real-valued independent identically distributed random variables $(X_{i})_{i=1}^n$ having densities. We study probability…
The Convex Minorant of the Cauchy Process
We determine the law of the convex minorant $(M_s, s\in [0,1])$ of a real-valued Cauchy process on the unit time interval, in terms of the gamma process. In particular, this enables us to deduce that…
Concave Majorants of Random Walks and Related Poisson Processes
We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst…
Sunset over Brownistan
...
...