EQUILIBRIUM, SYSTEMIC RISK MEASURES AND OPTIMAL TRANSPORT: A CONVEX DUALITY APPROACH

This Thesis focuses on two main topics. Firstly, we introduce and analyze the novel concept of Systemic Optimal Risk Transfer Equilibrium (SORTE), and we progressively generalize it (i) to a multivariate setup and (ii) to a dynamic (conditional) setting. Additionally we investigate its relation to a recently introduced concept of Systemic Risk Measures (SRM). We present Conditional Systemic Risk Measures and study their properties, dual representation and possible interpretations of the associated allocations as equilibria in the sense of SORTE. On a parallel line of work, we develop a duality for the Entropy Martingale Optimal Transport problem and provide applications to problems of nonlinear pricing-hedging. The mathematical techniques we exploit are mainly borrowed from functional and convex analysis, as well as probability theory. More specifically, apart from a wide range of classical results from functional analysis, we extensively rely on Fenchel-Moreau-Rockafellar type conjugacy results, Minimax Theorems, theory of Orlicz spaces, compactness results in the spirit of Komlós Theorem. At the same time, mathematical results concerning utility maximization theory (existence of optima for primal and dual problems, just to mention an example) and optimal transport theory are widely exploited. The notion of SORTE is inspired by the Bühlmann's classical Equilibrium Risk Exchange (H. Bühlmann, "The general economic premium principle", Astin Bulletin, 1984). In both the Bühlmann and the SORTE definition, each agent is behaving rationally by maximizing his/her expected utility given a budget constraint. The two approaches differ by the budget constraints. In Bühlmann's definition the vector that assigns the budget constraint is given a priori. In the SORTE approach, on the contrary, the budget constraint is endogenously determined by solving a systemic utility maximization problem. SORTE gives priority to the systemic aspects of the problem, in order to first optimize the overall systemic performance, rather than to individual rationality. Single agents' preferences are, however, taken into account by the presence of individual optimization problems. The two aspects are simultaneously considered via an optimization problem for a value function given by summation of single agents' utilities. After providing a financial and theoretical justification for this new idea, in this research sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of such a SORTE are presented. Once laid the theoretical foundation for the newly introduced SORTE, this Thesis proceeds in extending such a notion to the case when the value function to be optimized has two components, one being the sum of the single agents' utility functions, as in the aforementioned case of SORTE, the other consisting of a truly systemic component. This marks the progress from SORTE to Multivariate Systemic Optimal Risk Transfer Equilibrium (mSORTE). Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general setting allows us to introduce and study a Nash Equilibrium property of the optimizers. Existence, uniqueness, Pareto optimality and the Nash Equilibrium property of the newly defined mSORTE are proved in this Thesis. Additionally, it is shown how mSORTE is in fact a proper generalization, and covers both from the conceptual and the mathematical point of view the notion of SORTE. Proceeding further in the analysis, the relations between the concepts of mSORTE and SRM are investigated in this work. The notion of SRM we start from was introduced in the papers "A unified approach to systemic risk measures via acceptance sets" (Math. Finance, 2019) and "On fairness of systemic risk measures" (Finance Stoch., 2020) by F. Biagini, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis. SRM of Biagini et al. are generalized in this Thesis to a dynamic (namely conditional) setting, adding also a systemic, multivariate term in the threshold functions that Biagini et al. consider in their papers. The dynamic version of mSORTE is introduced, and it is proved that the optimal allocations of dynamic SRM, together with the corresponding fair pricing measures, yield a dynamic mSORTE. This in particular remains true if conditioning is taken with respect to the trivial sigma algebra, which is tantamount to working in the non-dynamic setting covered in Biagini et al. for SRM, and in the previous parts of our work for mSORTE. The case of exponential utility functions is thoroughly examined, and the explicit formulas we obtain for this specific choice of threshold functions allow for providing a time consistency property for allocations, dynamic SRM and dynamic mSORTE. The last part of this Thesis is devoted to a conceptually separate topic. Nonetheless, a clear mathematical link between the previous work and the one we are to describe is established by the use of common techniques. A duality between a novel Entropy Martingale Optimal Transport (EMOT) problem (D) and an associated optimization problem (P) is developed. In (D) the approach taken in Liero et al. (M. Liero, A. Mielke, and G. Savaré, "Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures", Inventiones mathematicae, 2018) serves as a basis for adding the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al.. The Problem (D) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive penalization terms D, which may not have a divergence formulation. In Problem (P) the objective functional, associated via Fenchel conjugacy to the terms D, is not any more linear, as in Optimal Transport or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a non linear subhedging value. Our results in this Thesis establish a novel nonlinear robust pricing-hedging duality in financial mathematics, which covers a wide range of known robust results in its generality. The research for this Thesis resulted in the production of the following works: F. Biagini, A. Doldi, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis, "Systemic optimal risk transfer equilibrium", Mathematics and Financial Economics, 2021; A. Doldi and M. Frittelli, "Multivariate Systemic Optimal Risk Transfer Equilibrium", Preprint: arXiv:1912.12226, 2019; A. Doldi and M. Frittelli, "Conditional Systemic Risk Measures", Preprint: arXiv:2010.11515, 2020; A. Doldi and M. Frittelli, "Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality", Preprint: arXiv:2005.12572, 2020.