47,612 research outputs found
Embedding laws in diffusions by functions of time
We present a constructive probabilistic proof of the fact that if
is standard Brownian motion started at , and is a
given probability measure on such that , then there
exists a unique left-continuous increasing function
and a unique left-continuous
decreasing function such
that stopped at or
has the law . The method of proof relies upon weak convergence arguments
arising from Helly's selection theorem and makes use of the L\'{e}vy metric
which appears to be novel in the context of embedding theorems. We show that
is minimal in the sense of Monroe so that the stopped process
satisfies natural uniform
integrability conditions expressed in terms of . We also show that
has the smallest truncated expectation among all stopping times
that embed into . The main results extend from standard Brownian
motion to all recurrent diffusion processes on the real line.Comment: Published at http://dx.doi.org/10.1214/14-AOP941 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An Optimal Skorokhod Embedding for Diffusions
Given a Brownian motion and a general target law (not necessarily
centered or even integrable) we show how to construct an embedding of in
. 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
. The embedding is then applied to regular diffusions, and used to
characterise the target laws for which a -embedding may be found.Comment: 22 pages, 4 figure
Classes of Skorokhod Embeddings for the Simple Symmetric Random Walk
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. Finally, we
define the natural extension of several known constructions from the Brownian
setting and show that these constructions require us to further restrict the
sets of target laws
On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale
We study the joint laws of a continuous, uniformly integrable martingale, its
maximum, and its minimum. In particular, we give explicit martingale
inequalities which provide upper and lower bounds on the joint exit
probabilities of a martingale, given its terminal law. Moreover, by
constructing explicit and novel solutions to the Skorokhod embedding problem,
we show that these bounds are tight. Together with previous results of Az\'ema
& Yor, Perkins, Jacka and Cox & Ob{\l}\'oj, this allows us to completely
characterise the upper and lower bounds on all possible exit/no-exit
probabilities, subject to a given terminal law of the martingale. In addition,
we determine some further properties of these bounds, considered as functions
of the maximum and minimum.Comment: 19 pages, 4 figures. This is the authors' accepted version of the
paper which will appear in Stochastic Processes and their Application
Optimal Transport and Skorokhod Embedding
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 optimal mass transport. In analogy to the celebrated article of Gangbo and
McCann on the geometry of optimal transport, we establish a geometric
characterization of Skorokhod embeddings with desired optimality properties.
This leads to a systematic method to construct optimal embeddings. It allows
us, for the first time, to derive all known optimal Skorokhod embeddings as
special cases of one unified construction and leads to a variety of new
embeddings. While previous constructions typically used particular properties
of Brownian motion, our approach applies to all sufficiently regular Markov
processes.Comment: Substantial revision to improve the readability of the pape
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