Boolean-valued models as a foundation for locally $L^0$-convex analysis and Conditional set theory

TL;DR

This paper demonstrates how Boolean-valued models can serve as a foundational framework for locally $L^0$-convex analysis and conditional set theory, enabling the transfer of classical results to these contexts, with applications in mathematical finance.

Contribution

It establishes a transfer principle using Boolean-valued models that allows classical theorems to be adapted to locally $L^0$-convex modules and conditional set theory, providing a new analytical approach.

Findings

Transfer of classical theorems to locally $L^0$-convex modules.

Boolean-valued models enable formulation of conditional set theory results.

Framework supports multi-period problems in mathematical finance.

Abstract

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of multi-period mathematical finance. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. By means of Boolean-valued models and its transfer principle we show that any known result on locally convex spaces has a transcription in the frame of locally $L^0$-convex modules which is also true, and that the formulation in conditional set theory of any theorem of classical set theory is also a theorem. We propose Boolean-valued analysis as an analytic framework for the study of multi-period problems in mathematical finance.