# Schedule of the Workshop "Optimal transport and stochastics"

## Monday, March 9

 14:30 - 15:15 Mete Soner: FTAP and robust duality 15:15 - 16:00 Rémi Lassalle: Variations of functionals for some semi-martingale optimal transportation problems 16:00 - 16:30 Coffee break 16:30 - 17:00 Bruno Bouchard: No-arbitrage under model ambiguity and Fundamental Theorems of Assert Pricing 17:00 - 17:30 Martin Larsson: Robust pricing by informed investors 17:30 - 18:00 Sigrid Källblad: Optimal Skorokhod embedding given full marginals and application to the maximal reward function

## Tuesday, March 10

 9:00 - 9:45 David Hobson: Integrability of solutions of the Skorokhod embedding problem for diffusions. 9:45 - 10:30 Marcel Nutz: Robust Finance and Dual Attainment in Martingale Optimal Transport 10:30 - 11:00 Coffee break 11:00 - 11:45 Jan Obloj: On robust pricing-hedging duality with pathspace restrictions in continuous time 11:45 - 12:30 Nizar Touzi: Azema-Yor and Root embeddings under multiple-marginals constraint, and Model-free hedging 12:30 - 14:15 Lunch break 14:15 - 14:45 Florian Stebegg: Root's Embedding and Kellerer's Theorem 14:45 - 15:15 Kostas Kardaras: Targeting wealth distributions under drawdown constraints 15:15 - 16:00 Claude Martini: Martingale measures with pre-specified marginals: extremal points and perturbations 16:00 - 16:30 Coffee break 16:30 - Discussion 19:00 - „Dinner at Machold“

## Wednesday, March 11

 9:00 - 9:45 Walter Schachermayer: Exponentially concave functions and multiplicative cyclical monotonicity 9:45 - 10:30 Peter Mörters: Optimal unbiased shifts of Brownian motion 10:30 - 11:00 Coffee break 11:00 - 11:45 Guillaume Carlier: A Monge-Kantorovich approach to multivariate quantile regression 11:45 - 12:30 Nicolas Juillet:  Martingale transport problem and PCOCs 12:30 - Lunch break, free afternoon 16:00 - 16:30 Coffee break

## Thursday, March 12

 9:00 - 9:45 Ali Üstünel: Some variational problems on the Wiener space and applications 9:45 - 10:30 Yan Dolinsky: Convex Duality with Transaction Costs 10:30 - 11:00 Coffee break 11:00 - 11:45 Xiaolu Tan: On the martingale optimal transport duality in the Skorokhod space 11:45 - 12:30 Mathias Beiglböck: The Geometry of Multi-Marginal Skorokhod Embedding 12:30 - 14:30 Lunch break 14:30 - 16:00 Discussion 16:00 - 16:30 Tea and cake

# Abstracts

## Mathias Beiglböck: The Geometry of Multi-Marginal Skorokhod Embedding

During the last 50 years the Skorokhod embedding problem has become an important classical problem in probability theory and a number of solutions with particular optimality properties have been constructed. Recently a unified derivation of many of these solutions has been obtained through a new approach inspired by the theory of optimal transport.

Using the original techniques from stochastic analysis, the multi-period version of the Skorokhod problem seems difficult and only limited results are available: Henry-Labordere, Obloj, Spoida, and Touzi derive the multi- marginal Azema-Yor embedding under additional technical conditions and recently the multi-marginal Root embedding has been obtained by Cox, Obloj, and Touzi.

Here we show that the transport approach can also be used to extend the classical optimal solutions to the multi-marginal Skorokhod problem. In particular we establish that these constructions share a common geometric structure. This has further applications to the martingale optimal transport problem.

(joint work with A. Cox, M. Huesmann)

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## Bruno Bouchard: No-arbitrage under model ambiguity and Fundamental Theorems of Assert Pricing

We will present several recent versions of the Fundamental Theorem of Asset Pricing for discrete and continuous time models under model ambiguity, with and without proportional transaction costs. This talks is based on recent collaborations with S. Biagini, K. Kardaras and M. Nutz.

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## Guillaume Carlier: A Monge-Kantorovich approach to multivariate quantile regression

The aim of this talk is to  present a way to extend the quantile regression method of Koenker and Bassett to the multivariate setting. A variant of the classical optimal transport problem with an additional mean-independence constraint plays a crucial role in the analysis. We shall also revisit the classical univariate case in the light of this variational point of view.  Joint work with Victor Chernozhukov (MIT) and Alfred Galichon (Sc. Po, Paris).

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## Yan Dolinsky: Convex Duality with Transaction Costs

Two different super-replication problems in a continuous time financial market with proportional transaction cost are considered. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are also allowed. The first one the problems considered is model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure $\{P\}$. The main result states that the two super-replication problems have the same value provided that $\{P\}$ satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost. Furthermore, a convex duality result is proved.

Joint work with H.M. Soner.

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## David Hobson: Integrability of solutions of the Skorokhod embedding problem for diffusions

Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq \R$ and let $\mu$ be a probability measure on $I^X$. Then $\tau$ is a solution of the Skorokhod embedding problem (SEP) for $\mu$ in $X$ if $\tau$ is a stopping time and $X_\tau \sim \mu$.
There are well-known conditions which determine whether there exists a solution of the SEP for $\mu$ 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 moment.
When $X$ is Brownian motion, there exists an integrable embedding of $\mu$ if and only if $\mu$ is centred and in $L^2$. Further, every integrable embedding is minimal. When $X$ is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If $Y$ is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify.

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## Nicolas Juillet: Martingale transport problem and PCOCs

We describe an approach to associate a continuous martingale $(M_t)_t$ to a PCOC, that is a family $(\mu_t)_t$ of real measures, where $t\mapsto \mu_t$ is increasing in the convex order. We want to satisfy the usual constraint, that, for every $t$ the law of $M_t$ is $\mu_t$. The approach relies on the martingale version of the optimal transport theory. We discuss several cases and issues.

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## Sigrid Källblad: Optimal Skorokhod embedding given full marginals and application to the maximal reward function

We consider here the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0,1]. The problem is related to studying the extremal martingales associated to a peacock (“process increasing in convex ordering”, by Hirsch, Profeta, Roynette and Yor (2011)). A general duality result is obtained by convergence techniques. We then study the case where the reward functions depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordere, Obloj, Spoida and Touzi (2014). Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality. This is joint work with Xiaolu Tan and Nizar Touzi.

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## Kostas Kardaras: Targeting wealth distributions under drawdown constraints

In a continuous-path semimartingale model, we consider an arbitrary distribution on the positive real line representing desired log-returns on investment. Under drawdown constraints, we explicitly obtain a wealth process and a stopping time that achieve the previous distribution.

Importantly, the wealth process involves a mutual fund in the cash account and the growth-optimal wealth process in a model-independent way (as long as these two funds are provided by the market). The stopping rule depends only on directly observable quantities and, of course, the target distribution. Certain optimality properties of the given solution will be discussed.

(Based on ongoing research with J. Obloj, E. Platen and N. Rhodosthenous.)

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## Martin Larsson: Robust pricing by informed investors

Well-informed agents can hedge more efficiently than poorly informed agents, even if they have access to the same set of traded securities. In a robust framework with semi-static trading opportunities, we study super-hedging prices obtained by agents with different filtrations. Under structural assumptions, we find that informed agents compute super-hedging prices using only those probability measures that render the additional information inconsequential. The theory of filtration enlargement plays an important role, as well as the notion of semi-static completeness. This is joint work with Beatrice Acciaio.

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## Rémi Lassalle: Variations of functionals for some semi-martingale optimal transportation problems

In this talk I will introduce some variations processes, which allow us to differentiate some functionals depending on the characteristics of laws of some continuous semi-martingales. First I will give the motivations and add some remarks on connections and differences between some semi-martingale optimal transport problems and random mechanics. Then I will point out the technical difficulties which arise in this particular framework (i.e. to differentiate functionals depending on laws and on their characteristics), and I will show how these difficulties lead us naturally to introduce a class of variation processes. Then I will give some general results, which are then applied to some semi-martingale optimal transportation problems.

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## Claude Martini: Martingale measures with pre-specified marginals: extremal points and perturbations

The extremal points in the set of all measures with pre-specified marginals, without the martingale constraint, have been extensively studied by many authors in the past (e.g. Denny, Douglas, Letac, Klopotowski to cite only a few). In this talk, we will focus on the characterization provided by Denny in the countable case: a key property is that the support of the probability Q has no “cycle”, otherwise a perturbation of Q can be constructed so that Q can not be extremal. In the context of the 2 marginals martingale problem studied by Beiglböck-Juillet, with special cases provided by Henry-Labordère and Touzi, Hobson and Klimmeck, Hobson and Neuberger, and Laachir, we give an analogous "cycle-like" property that corresponds to martingale perturbations.

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## Peter Mörters: Optimal unbiased shifts of Brownian motion

For two-sided Brownian motion $\{B_t : t\in R\}$ an unbiased shift is a random time $T$ such that $\{B_{t+T} : t\in R\}$ is again a two-sided Brownian motion. Given two orthogonal probability measures $\mu$ and $\nu$, such that $B_0$ has distribution $\mu$, we explicitly construct an unbiashed shift $T$, which is also a stopping time, such that $B_T$ has distribution $\nu$. We show that this solution simultaneously minimises the expectation of all concave functions of $T$ among all nonnegative solutions of the embedding problem for unbiased shifts. The talk is based on joint work with Guenter Last (Karlsruhe), Hermann Thorisson (Reykjavik) and Istvan Redl (Bath).

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## Marcel Nutz: Robust Finance and Dual Attainment in Martingale Optimal Transport

We study optimal static hedging with options in the context of model-free finance and the implications for the so-called martingale optimal transport problem. It turns out that the existence of optimal options is related to a quasi-sure concept of market viability. Based on joint works with M. Beiglböck, B. Bouchard and N. Touzi.

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## Jan Obloj: On robust pricing-hedging duality with pathspace restrictions in continuous time

In this talk I show how to interpolate between the two worlds. I argue that quoted option prices should be incorporated through distributional constraints while beliefs, or past data, are most naturally included through pathwise restrictions. The resulting framework is robust and flexible, allowing for realistic outputs while quantifying the impact of making assumptions.
I will focus on abstract results about pricing-hedging duality. They correspond to a version of optimal transport duality with transport along paths of a martingale with constrained support and with constrained but not necessarily fully specified target distribution.
The talk is based on joint work with Zhaoxu Hou (University of Oxford).

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## Walter Schachermayer: Exponentially concave functions and multiplicative cyclical monotonicity

We present some recent work by S. Pal and T. Wong relating stochastic portfolio theory with a multiplicative version of the notion of cyclical monotonicity.

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## Mete Soner: FTAP and robust duality

In non-dominated financial markets the notion of arbitrage is non trivial.  In the quasi-sure setting, Bouchard & Nutz consider the probabilistic extension of the classical definition and prove a deep characterization of the FTAP in that context. Acciaio et.al.  considers the model-free approach in which the notion of arbitrage is different and prove the FTAP for that definition.  Burzoni et.al. provides a general framework which allows several for different notions.  In related research, Monge-Kantorovich type duality between super-replication and "equivalent" martingale measures have been proved for several markets.  As discussed in a joint paper of mine with Dolinsky, these results are also closely connected and one can prove some versions of FTAP from duality.  In this talk, I will outline these results.

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## Florian Stebegg: Root's Embedding and Kellerer's Theorem

We observe that Root's solution of the Skorokhod embedding problem gives rise to martingale transport plans which enjoy a particular Lipschitz-property. Using compactness of the set of all martingale measures with prescribed marginals, Kellerer's classical theorem is obtained.

(with M. Beiglböck, M. Huesmann)

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## Xiaolu Tan: On the martingale optimal transport duality in the Skorokhod space

We study a martingale optimal transport problem in the Skorokhod space of cadlag paths, under finitely or infinitely many marginals constraint. To establish a general duality result, we utilize a Wasserstein type topology on the space of measures on the real value space, and the S-topology introduced by Jakubowski (1997) on the Skorokhod space, together with the discretization technique in Dolinsky and Soner (2014).

This is a joint work with Gaoyue Guo and Nizar Touzi.

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## Nizar Touzi: Azema-Yor and Root embeddings under multiple-marginals constraint, and Model-free hedging

We provide solutions to the problems of model-free superhedging of Lookback options and Variance options under finitely many marginals constraints. These solutions correspond to a finitely-many marginals version of the Azema-Yor and the Root solutions of the Skorohod embedding problem.

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