DE MARCH Hadrien

In this paper we propose a mathematical framework to address the uncertainty emergingwhen the designer of a trading algorithm uses a threshold on a signal as a control. We rely ona theorem by Benveniste and Priouret to deduce our Inventory Asymptotic Behaviour (IAB)Theorem giving the full distribution of the inventory at any point in time for a well formulatedtime continuous version of the trading algorithm.Since this is the first time a paper proposes to address the uncertainty linked to the use of athreshold on a signal for trading, we give some structural elements about the kind of signals thatare using in execution. Then we show how to control this uncertainty for a given cost function.There is no closed form solution to this control, hence we propose several approximation schemesand compare their performances.Moreover, we explain how to apply the IAB Theorem to any trading algorithm drivenby a trading speed. It is not needed to control the uncertainty due to the thresholding of asignal to exploit the IAB Theorem. it can be applied ex-post to any traditional trading algorithm.

Martingale transport plans on the line are known from Beiglbock & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d larger than one. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we check that this algorithm performs better through numerical experiments. We also give a simple way to deal with the absence of convex ordering among the marginals. Furthermore, we provide a new universal bound on the error linked to entropy.

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 Beiglb\"ock, 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 3 provided that the target measure is dominated by the Lebesgue measure. More generally, our results hold in any dimension under an assumption which is implied by the Continuum Hypothesis. Finally, in contrast with the one-dimensional setting of Beiglb\"ock, Lim & Obloj, we provide an example which illustrates that the smoothness of the coupling function does not imply that pointwise duality holds for compactly supported measures.

We consider the classical problem of building an arbitrage-free implied volatility surface from bid-ask quotes. We design a fast numerical procedure, for which we prove the convergence, based on the Sinkhorn algorithm that has been recently used to solve efficiently (martingale) optimal transport problems.

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 Lipschitz couplings. Using tools from algebraic geometry, we provide sufficient conditions for finiteness of this conditional support, together with (optimal) lower bounds on the maximal cardinality for a given coupling function. More results are obtained for specific examples of coupling functions based on distance functions. In particular, we show that the above conjecture of Ghoussoub, Kim & Lim is not valid beyond the context of atomic target distributions.

In this thesis we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally propose methods of numerical approximation.We first prove the existence of irreducible components intrinsic to martingale transports between two given measures, and the canonicity of these components. We then prove a duality result for the optimal martingale transport in any dimension, the point by point duality is no longer true but a form of quasi-safe duality is proved. This duality allows us to prove the possibility of decomposing the quasi-safe optimal transport into a series of subproblems of point by point optimal transports on each irreducible component. We finally use this duality to prove a martingale monotonicity principle, analogous to the famous monotonicity principle of classical optimal transport. We then study the local structure of optimal transports, deduced from differential considerations. We obtain a characterization of this structure using real algebraic geometry tools. We deduce the structure of martingale optimal transports in the case of Euclidean norm power costs, thus solving a conjecture dating back to 2015. Finally, we compare existing numerical methods and propose a new method that is shown to be more efficient and to deal with an intrinsic problem of the martingale constraint that is the convex order defect. We also give techniques to handle the numerical problems in practice.