Optimal Transport: A Semi-Discrete Approach

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? 
If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. 
Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

Adapted Wasserstein Distances and Stability in Mathematical Finance

Assume that an agent models a financial asset through a measure ℚ with the goal to price/hedge some derivative or optimise some expected utility. Even if the model ℚ is chosen in the most skilful and sophisticated way, the agent is left with the possibility that ℚ does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of ℚ? If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

/pdf/adapted-wasserstein-distances-and-stability-in-mathematical-1c82e5l40n.pdf

Adapted Wasserstein distances and stability in mathematical finance

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on $$\mathbb {R}^d$$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

/pdf/existence-duality-and-cyclical-monotonicity-for-weak-4ow9axrsug.pdf

Existence, duality, and cyclical monotonicity for weak transport costs

Motivated by the approximation of Martingale Optimal Transport problems, we
study sampling methods preserving the convex order for two probability measures
$\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When
$(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$
(resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convex
order. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than
$\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly converging
to $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertz
and R\"osler (1992), the set of probability measures with a finite first order
moment is a lattice for the increasing and the decreasing convex orders. From
this result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greater
than $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficient
algorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and
$\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$
and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection
$\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) of $\mu$ (resp. $\nu$)
on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$)
in the convex order for the Wasserstein distance with index $\rho$. When
$\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving a
quadratic optimization problem with linear constraints. It turns out that, in
dimension 1, the projections do not depend on $\rho$ and their quantile
functions are explicit, which leads to efficient algorithms for convex
combinations of Dirac masses. Last, we illustrate by numerical experiments the
resulting sampling methods that preserve the convex order and their application
to approximate Martingale Optimal Transport problems.

Sampling of probability measures in the convex order by Wasserstein projection

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ 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 optimal transport plans $\pi$ are not characterized by a `monotonicity'-property of their support. 
In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet. An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.

Stability of martingale optimal transport and weak optimal transport.

The optimal weak transport problem has recently been introduced by Gozlan et.\ al. We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier-Strassen Theorem of Gozlan-Juillet to general probability measures on $R^d$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (`adapted') topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

Existence, Duality, and Cyclical monotonicity for weak transport costs

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ 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 optimal transport plans $\pi$ are not characterized by a `monotonicity'-property of their support. In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet. An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.

Stability of martingale optimal transport and weak optimal transport

Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali introduced a transport problem for `weak' cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality. In this article we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schr\"odinger problem, the Brenier-Strassen theorem, optimal mechanism design, linear transfers and semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

Gudmund Pammer

Papers

Existence, duality, and cyclical monotonicity for weak transport costs

Stability of martingale optimal transport and weak optimal transport.

Existence, Duality, and Cyclical monotonicity for weak transport costs

Stability of martingale optimal transport and weak optimal transport

Applications of weak transport theory