A Boolean-valued Models Approach to L-Convex Analysis, Conditional Risk and Stochastic Control

Abstract

Boolean-valued analysis is a branch of functional analysis, which consists in studying the properties of mathematical objects by means of comparison between its representations in different set-theoretic models whose construction uses distinct Boolean algebras. Boolean-valued analysis stems from the formalization given by Scott, R. Solovay, and P. Vopenka of the method of forcing that Paul Cohen created to prove the independence of the continunm hypothesis. Duality theory of risk measures is a branch of mathematical finance. In particular, duality theory of conditional risk measures studies the case in which a dynamic flow of information throughout the time is taken into account. The difficulties of this problem have motivated new developments in functional analysis such as L0-convex analysis and conditional set theory. This work aims: 3. To show that both L0-convex analysis and conditional set theory are particular instances of Boolean-valued analysis and use the tools of a well-known and deep mathematical theory. 4. To look for applications to problems of mathematical finance. Results and methodology: In Chapter 1, by means of a miscellaneous of results and boundary examples of L0-modules, it is provided a study of algebraic and topological properties of L0-modules. In Chapter 2, it is proven that L0-convex analysis is a Boolean-valued interpretation of classical convex analysis. This allows us to transcribe classical theorem of convex analysis as new theorems of L0-convex analysis, which hold without the necessity of a proof. For instance, we provide versions of the Brouwer's fixed point theorem, of the Eberlein-Smulian theorem, of the Krein-Smulian theorem, of the Mazur theorem and of the James' compactness theorem as well as a perturbed version of the latter. In Chapter 3, it is shown that conditional set theory is a Boolean-valued interpretation of classical set theory. As instances of application, we provide conditional versions of classical theorems of functional analysis, which hold without the necessity of a proof. In Chapter 4, it is shown that duality theory of conditional risk measures is a Boolean-valued interpretation of duality theory of conventional risk measures. As a consequence, it is obtained that each theorem on dual representation of conventional risk measures can be interpreted as a new theorem on dual representation of conditional risk measures. As application, it is established a general robust representation theorem of conditional risk measures. It is studied the form of this theorem in the cases of Lp type modules, Orlicz type modules, and Orlicz-heart type modules. In Chapter 5, by applying L0-convex analysis, parameter-dependent stochastic optimization in finite time is studied. Two theorems on the existence of optimal solutions of the stochastic control problem are proven: one under compactness conditions on the controls and the other without constraints on the controls but under stronger assumptions on the forward and backward generators. These results are then applied to specific problems of utility maximization and sharing. By means of these techniques, a new proof of Dalang-Morton-Willinger theorem is provided. Conclusions: It is shown that both L0-convex anaysis and conditional set theory are particular instances of Boolean-valued analysis. In particular, the transfer principle allows us to establish a module analogue of classical convex analysis. Furthermore, we conclude that these tools can satisfactorily be applied to conditional risk and stochastic control.