In this paper we shall prove that the plane of financial events, intro- duced and applied to financial problems by the author himself can be considered as a bration in two different ways. The first one, the natural one, reveals itself to be isomorph to the tangent- bundle of the real line, when the last one is considered as a differentiable manifold in the natural way; the second one is a fibration induced by the status of compound interest capitalization at a given rate. Moreover, in the paper we dene on the first bration an ane connec- tion, also in this case induced by the status of compound interest at a given rate i. The final goal of this paper is the awareness that all the effects determined by the status of compound interest are nothing but the consequences of the fact that the space of financial events is a fibration endowed with a particular affine connection, so they are consequences of purely geometric properties, at last, depending upon the curvature deter- mined by the connection upon the fibration. A natural preorder upon the set of fibers of the second fibration is considered. Some remarks about the applicability to economics and finance of the theories presented in the paper and about the possible developements are made in the directions followed in papers of the author. We see also the evolution of a capitalized financial event e, with respect to a capitaliza- tion factor f, as the exponential map of a suitably defined Lie group Gf;e, supported by the halfspace of capitalized financial events having the same sign of e. The Lie group Gf;e depends on the capitalization factor f and on the event e itself. After the extention of the definition of exponential map of a Lie group, we shall eliminate the dependence on the financial event e, recognizing the precence of essentialy one unique financial Lie semigroup, supported by the entire space of capitalized financial events, determined by the capitalization factor f.

Fibred spaces and financial structures

CARFI', David
2011

Abstract

In this paper we shall prove that the plane of financial events, intro- duced and applied to financial problems by the author himself can be considered as a bration in two different ways. The first one, the natural one, reveals itself to be isomorph to the tangent- bundle of the real line, when the last one is considered as a differentiable manifold in the natural way; the second one is a fibration induced by the status of compound interest capitalization at a given rate. Moreover, in the paper we dene on the first bration an ane connec- tion, also in this case induced by the status of compound interest at a given rate i. The final goal of this paper is the awareness that all the effects determined by the status of compound interest are nothing but the consequences of the fact that the space of financial events is a fibration endowed with a particular affine connection, so they are consequences of purely geometric properties, at last, depending upon the curvature deter- mined by the connection upon the fibration. A natural preorder upon the set of fibers of the second fibration is considered. Some remarks about the applicability to economics and finance of the theories presented in the paper and about the possible developements are made in the directions followed in papers of the author. We see also the evolution of a capitalized financial event e, with respect to a capitaliza- tion factor f, as the exponential map of a suitably defined Lie group Gf;e, supported by the halfspace of capitalized financial events having the same sign of e. The Lie group Gf;e depends on the capitalization factor f and on the event e itself. After the extention of the definition of exponential map of a Lie group, we shall eliminate the dependence on the financial event e, recognizing the precence of essentialy one unique financial Lie semigroup, supported by the entire space of capitalized financial events, determined by the capitalization factor f.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/1912811
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