Financial Lie groups

TL;DR

This paper models the evolution of capitalized financial events using Lie group theory, introducing a new mathematical framework that captures financial dynamics through exponential maps and semigroups.

Contribution

It develops a novel Lie group and semigroup framework for financial events, extending exponential maps to describe financial evolution independently of specific events.

Findings

Defines a financial Lie group G(f,e) based on capitalization factors and events

Extends exponential map concepts to financial contexts

Identifies a unique financial Lie semigroup for all capitalized events

Abstract

In this paper we see the evolution of a capitalized financial event e, with respect to a capitalization factor f, as the exponential map of a suitably defined Lie group G(f,e), supported by the half-space of capitalized financial events having the same capital sign of e. The Lie group G(f,e) depends upon the capitalization factor f and on the event e itself. After the extension of the definition of exponential map of a Lie group, we shall eliminate the dependence on the financial event e, recognizing the presence of essentially one unique financial Lie semigroup, supported by the entire space of capitalized financial events, determined by the capitalization factor f.