A classical result of Strassen asserts that given probabilities $μ, ν$ on the real line which are in convex order, there exists a \emph{martingale coupling} with these marginals, i.e.\ a random vector $(X_1,X_2)$ such that $X_1\sim μ, X_2\sim ν$ and $E[X_2|X_1]=X_1$. Remarkably, it is a non trivial problem to construct particular solutions to this problem. In this article, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of optimality properties / geometry of the support set / representation through a Skorokhod embedding. As a particular element of this family we recover the (left-) curtain martingale transport, which has recently been studied \cite{BeJu16, HeTo13, CaLaMa14, BeHeTo15} and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usual \emph{product coupling} and appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addition, this coupling provides an explicit example of a Lipschitz-kernel, shedding new light on Kellerer's proof of the existence of Markov martingales with specified marginals.
Given a family of real probability measures $(μ_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(ν^α)_{α\in [0,1]}$ with $ν^α(\mathbb{R})=α$ that are submeasures of $μ_0$, called a parametrization of $μ_0$. Then, for any $α$ we define an evolution $(η^α_t)_{t\geq 0}$ of the …
It is well known that given two probability measures $μ$ and $ν$ on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow measure which induces a family of martingale couplings, and solv…
The (left-)curtain coupling, introduced by Beiglböck and the author is an extreme element of the set of "martingale" couplings between two real probability measures in convex order. It enjoys remarkable properties with respect to order relations and a minimisation problem inspired by the theory of optimal transport. An explicit representation and a number of further noteworthy attributes have recently been established by Henry-Labordère and Touzi. In the present paper we prove that the curtain coupling depends continuously on the prescribed marginals and quantify this with Lipschitz estimates.…
For two measures $μ$ and $ν$ that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab., 46(6):3351--3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling $π^D$ by establishing a Brenier-type result: (a generalised version of) $π^D$ concentrates on the graphs of two functions. Our constructio…
The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351--3398, 2018) is an extreme point of the set of `supermartingale' couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to…
We are interested in martingale rearrangement couplings. As introduced by Wiesel [37] in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the orig…
In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $μ$ and $ν$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $μ$ and $ν$. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of $\vert x-y\vert$ with respect to each of these couplings is smaller th…
Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale …
Beiglböck and Juillet ("On a problem of optimal transport under marginal martingale constraints") introduced the left-curtain martingale coupling of probability measures $μ$ and $ν$, and proved that, when the initial law $μ$ is continuous, it is supported by the graphs of two functions. We extend the later result by constructing the generalised left-curtain martingale coupling and show that for an arbitrary starting law $μ$ it is characterised by two appropriately defined lower and upper functions. As an application of this result we derive the model-independent upper bound of an American put …
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove th…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal marti…
We show continuity of the martingale optimal transport optimisation problem as a functional of its marginals. This is achieved via an estimate on the projection in the nested/causal Wasserstein distance of an arbitrary coupling on to the set of martingale couplings with the same marginals. As a corollary we obtain an independent proof of sufficiency of the monotonicity principle established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab., 44 (2016), no. 1, 42106] for cost functions of polynomial growth.
In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of …
We show an intimate connection between solutions of the Skorokhod Embedding Problem which are given as the first hitting time of a barrier and the concept of shadows in martingale optimal transport. More precisely, we show that a solution $τ$ to the Skorokhod Embedding Problem between $μ$ and $ν$ is of the form $τ= \inf \{t \geq 0 : (X_t,B_t) \in \mathcal{R}\}$ for some increasing process $(X_t)_{t \geq 0}$ and a barrier $\mathcal{R}$ if and only if there exists a time-change $(T_l)_{l \geq 0}$ such that for all $l \geq 0$ the equation $$\mathbb{P}[B_τ \in \cdot , τ\geq T_l] = \mathcal{S}^ν(\m…
In a martingale optimal transport (MOT) problem mass distributed according to the law $μ$ is transported to the law $ν$ in such a way that the martingale property is respected. Beiglböck and Juillet (On a problem of optimal transport under marginal martingale constraints, Annals of Probability, 44(1):42-106, 2016) introduced a solution to the MOT problem which they baptised the left-curtain coupling. The left-curtain coupling has been widely studied and shown to have many applications, including to martingale inequalities and the model-independent pricing of American options. Beiglböck and Jui…
This paper analyzes the support of the conditional distribution of optimal martingale transport plans in higher dimension. In the context of a distance coupling in dimension larger than 2, previous results established by Ghoussoub, Kim & Lim show that this conditional transport is concentrated on its own Choquet boundary. Moreover, when the target measure is atomic, they prove that the support is concentrated on d+1 points, and conjecture that this result is valid for arbitrary target measure. We provide a structure result of the support of the conditional optimal transport for general Lipschi…
The paper studies a non-linear transformation between Brownian martingales, which is given by the inverse of the pricing operator in the mathematical finance terminology. Subsequently, the solvability of systems of equations corresponding to such transformations is investigated. The latter give rise to novel monotone pathwise couplings of an arbitrary number of certain diffusion processes with varying diffusion coefficients. In the case that there is an uncountable number of these diffusion processes and that the index set is an interval such couplings can be viewed as models for the growth of…
This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales. The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The …
This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X,Λ)$. The process $(X,Λ)$, called a regime-switching jump diffusion with Lévy type jumps, is further shown to posses Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.
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.
Hoeffding has shown that tail bounds on the distribution for sampling from a finite population with replacement also apply to the corresponding cases of sampling without replacement. (A special case of this result is that binomial tail bounds apply to the corresponding hypergeometric tails.) We give a new proof of Hoeffding's result by constructing a martingale coupling between the sampling distributions. This construction is given by an explicit combinatorial procedure involving balls and urns. We then apply this construction to create martingale couplings between other pairs of sampling dist…
Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglböck, Nutz & Touzi. Similar, we also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. As another contribution, we extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, and…
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.
We revisit Kellerer's Theorem, that is, we show that for a family of real probability distributions $(μ_t)_{t\in [0,1]}$ which increases in convex order there exists a Markov martingale $(S_t)_{t\in[0,1]}$ s.t.\ $S_t\sim μ_t$. To establish the result, we observe that the set of martingale measures with given marginals carries a natural compact Polish topology. Based on a particular property of the martingale coupling associated to Root's embedding this allows for a relatively concise proof of Kellerer's theorem. We emphasize that many of our arguments are borrowed from Kellerer \cite{Ke72}, Lo…
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimension $d$ is arbitrary.
The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb {E}[c(X_1,X_2)]$ by varying the joint distribution $(X_1,X_2)$ where the marginal distributions of the random variables $X_1$ and $X_2$ are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that $(X_i)_{i=1,2}$ is a martingale, that is, $\mathbb {E}[X_2|X_1]=X_1$. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific …
We consider a coupling of the Gaussian free field with slit holomorphic stochastic flows, called ($δ,σ$)-SLE, which contains known SLE processes (chordal, radial, and dipolar) as particular cases. In physical terms, we study a free boundary conformal field theory with one scalar bosonic field, where Green's function is assumed to have some general regular harmonic part. We establish which of these models allow coupling with ($δ,σ$)-SLE, or equivalently, when the correlation functions induce local ($δ,σ$)-SLE martingales (martingale observables).
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $π_1, π_2, \ldots$ converges weakly to a transport plan $π$, then $π$ is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since opti…
In this paper we apply change of numeraire techniques to the optimal transport approach for computing model-free prices of derivatives in a two periods model. In particular, we consider the optimal transport plan constructed in \cite{HobsonKlimmek2013} as well as the one introduced in \cite{BeiglJuil} and further studied in \cite{BrenierMartingale}. We show that, in the case of positive martingales, a suitable change of numeraire applied to \cite{HobsonKlimmek2013} exchanges forward start straddles of type I and type II, so that the optimal transport plan in the subhedging problems is the same…
Consider a multiperiod optimal transport problem where distributions $μ_{0},\dots,μ_{n}$ are prescribed and a transport corresponds to a scalar martingale $X$ with marginals $X_{t}\simμ_{t}$. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence--Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if $μ_{0}$ is atomless, but not in general. In the one-period case …
We establish a central limit theorem for (a sequence of) multivariate martingales which dimension potentially grows with the length $n$ of the martingale. A consequence of the results are Gaussian couplings and a multiplier bootstrap for the maximum of a multivariate martingale whose dimensionality $d$ can be as large as $e^{n^c}$ for some $c>0$. We also develop new anti-concentration bounds for the maximum component of a high-dimensional Gaussian vector, which we believe is of independent interest. The results are applicable to a variety of settings. We fully develop its use to the estimation…
Two probability distributions $μ$ and $ν$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cos…
