Skorokhod Embedding Research Articles

Minimal Root’s embeddings for general starting and target distributions

Most results regarding Skorokhod embedding problems SEP so far rely on the assumption that the corresponding stopped process is uniformly integrable, which is equivalent to the convex ordering condition Uμ≤Uν when the underlying process is a local martingale. In this paper, we study the existence, construction of Root’s solutions to SEP, in the absence of this convex ordering condition. We replace the uniform integrability condition by the minimality condition (Monroe, 1972), as the criterion of “good” solutions. A sufficient and necessary condition (in terms of local time) for minimality is given. We also discuss the optimality of such minimal solutions. These results extend the generality of the results given by Cox and Wang (2013) and Gassiat et al. (2015). At last, we extend all the results above to multi-marginal embedding problems based on the work of Cox et al. (2018).

Robust hedging of options on a leveraged exchange traded fund

A leveraged exchange traded fund (LETF) is an exchange traded fund that uses financial derivatives to amplify the price changes of a basket of goods. In this paper, we consider the robust hedging of European options on a LETF, finding model-free bounds on the price of these options. To obtain an upper bound, we establish a new optimal solution to the Skorokhod embedding problem (SEP) using methods introduced in Beiglbock–Cox–Huesmann. This stopping time can be represented as the hitting time of some region by a Brownian motion, but unlike other solutions of, for example, Root, this region is not unique. Much of this paper is dedicated to characterising the choice of the embedding region that gives the required optimality property. Notably, this appears to be the first solution to the SEP where the solution is not uniquely characterised by its geometric structure, and an additional condition is needed on the stopping region to guarantee that it is the optimiser. An important part of determining the optimal region is identifying the correct form of the dual solution, which has a financial interpretation as a model-independent superhedging strategy.

A conformal Skorokhod embedding

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.

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.

From optimal stopping boundaries to Rost’s reversed barriers and the Skorokhod embedding

We investigate the connection between Rost’s solution of the Skorokhod embedding problem and a suitable family of optimal stopping problems for Brownian motion with nite time-horizon. In particular we prove by probabilistic methods and stochastic calculus that the time reversal of the optimal stopping sets for such problems form the so-called Rost’s reversed barrier. MSC2010: 60G40, 60J65, 60J55, 35R35.

Existence of Transport Plans with Domain Constraints

Let $\Omega$ be one of $\X^{N 1},C[0,1],D[0,1]$: product of Polish spaces, space of continuous functions from $[0,1]$ to $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$, respectively. We first consider the existence of a probability measure $P$ on $\Omega$ such that $P$ has the given marginals $\alpha$ and $\beta$ and its disintegration $P_x$ must be in some fixed $\Gamma(x) \subset \kP(\Omega)$, where $\kP(\Omega)$ is the set of probability measures on $\Omega$. The main application we have in mind is the martingale optimal transport problem when the martingales are assumed to have bounded volatility/quadratic variation. We show that such probability measure exists if and only if the $\alpha$ average of the so-called $G$-expectation of bounded continuous functions with respect to the measures in $\Gamma$ is less than their $\beta$ average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer.

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.

Numerical scheme for Dynkin games under model uncertainty

We introduce an efficient numerical scheme for continuous time Dynkin games under model uncertainty. We use the Skorokhod embedding in order to construct recombining tree approximations. This technique allows us to determine convergence rates and to construct numerically optimal stopping strategies. We apply our method to several examples of game options.

Recombining Tree Approximations for Optimal Stopping for Diffusions

In this paper we develop two numerical methods for optimal stopping in the framework of one dimensional diffusion. Both of the methods use the Skorokhod embedding in order to construct recombining ...

Integrated quantile functions: properties and applications

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon--Walsh construction in the Skorokhod embedding problem.

Some Results on Skorokhod Embedding and Robust Hedging with Local Time

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 a generalization of the Vallois solution. It turns out from our analysis that one needs to relax the monotonicity assumption on the embedding functions in order to embed a larger class of marginal distributions. Finally, in a full-marginal setting where the stopping times given by Vallois are well ordered, we construct a remarkable Markov martingale which provides a new example of fake Brownian motion.

Pathwise superreplication via Vovk\u2019s outer measure

Since Hobson’s seminal paper (Hobson in Finance Stoch. 2:329–347, 1998), the connection between model-independent pricing and the Skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing–hedging duality for financial derivatives which are susceptible to the Skorokhod approach.Using Vovk’s approach to mathematical finance, we derive a model-independent superreplication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

Monotonicity preserving transformations of MOT and SEP

Recently, Beiglböck and Juillet (2016) and Beiglböck et al. (2015) established that optimizers to the martingale optimal transport problem (MOT) are concentrated on c-monotone sets. In this article we characterize monotonicity preserving transformations revealing certain symmetries between optimizers of MOT for different cost functions. Due to the intimate connection of MOT and the Skorokhod embedding problem (SEP) these transformations are also monotonicity preserving and disclose symmetries for certain solutions to the optimal SEP. Furthermore, the SEP picture allows to easily understand the geometry of these transformations once we have established the SEP counterparts to the known solutions of MOT based on the monotonicity principle for SEP which in turn allows to directly read off the structure of the MOT optimizers.

An iterated Azéma–Yor type embedding for finitely many marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azema and Yor [In Seminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordere et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

Stopping with expectation constraints: 3 points suffice

We consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.

Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509–536] peacock under their “increasing mean residual value” condition. We also discuss the associated martingale inequality.

Integral equations for Rost’s reversed barriers: Existence and uniqueness results

We establish that the boundaries of the so-called Rost’s reversed barrier are the unique couple of left-continuous monotonic functions solving a suitable system of nonlinear integral equations of Volterra type. Our result holds for atom-less target distributions μ of the related Skorokhod embedding problem.The integral equations we obtain here generalise the ones often arising in optimal stopping literature and our proof of the uniqueness of the solution goes beyond the existing results in the field.

Monotone martingale transport plans and Skorokhod embedding

We show that the left-monotone martingale coupling is optimal for any given performance function satisfying the martingale version of the Spence–Mirrlees condition, without assuming additional structural conditions on the marginals. We also give a new interpretation of the left monotone coupling in terms of Skorokhod embedding which allows us to give a short proof of uniqueness.

Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

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.