THE FAMILY OF OPERATORS ASSOCIATED WITH A CAPITALIZATION LAW

ABSTRACT. In the present paper we conduct a deep study on some basic concepts of fi-nancial mathematics. The financial mathematics involves several mathematical tools (even though at elementary level). But these tools are not organized in a theoretical framework. This circumstance affected badly on the development of financial mathematics and its ap-plications. The aim of this paper is to present the building of a theoretical structure for the basic part of financial mathematics, all with elementary mathematics. In particular, the goal consists of describing the “evolution ” of an amount of money as a time dependent-process, using the one parameter families of operators. The generalization to the case of the evolution of a portfolio of goods is very natural. In section 1, we define the capitalizations and the family of the capitalization factors of a capitalization. In section 2, we give a definition of capitalization law, more general than the usual one. Furthermore, we introduce the main concept of the paper: the family of operators of a capitalization law. In section 3, we connect the financial laws to the capitalizations. In section 4, we introduce a subclass of capitalization laws (the self-actualizing capi-talization laws). Finally, we characterize them and the classic separable financial laws in the new framework. We have to note that almost all the proofs of the paper are straightforward computa-tions. Nevertheless, we resolved to give them, we did it not so many for desire of com-pletness, but, above all, to build a bridge beetwen the language and thoughts of economists and mathematicians. 1. Capitalizations on compact intervals We bigin formalizing the basic object of financial mathematics, i.e., the financial evolu-tion of a capital (or a debt). Definition 1.1 (of capitalization on a compact interval). Let be an interval of the real line. We define capitalization (resp. debt-evolution) o