Justificación de la Necesidad de Utilizar el Cálculo Estocástico en Financiera

  1. García Villalón, Julio 1
  2. Rodríguez Ruiz, Julián 2
  3. Seijas, J. Antonio 3
  1. 1 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

  2. 2 Universidad Nacional de Educación a Distancia
    info

    Universidad Nacional de Educación a Distancia

    Madrid, España

    ROR https://ror.org/02msb5n36

  3. 3 Universidade da Coru˜na
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Journal:
Anales de ASEPUMA

ISSN: 2171-892X

Year of publication: 2015

Issue: 23

Type: Article

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Abstract

The job starts, giving a concise idea of the theory of Lebesgue integration, Stieltjes and convergence theorems. A reference to ”versus Stochastic Calculus Cal-culus Classic” is made, noting that the classical relationship is no longer applicable for real functions which are frequently presented in Financial Mathematics. In the nineteenth century, the German mathematician WEIERSTRAS Karl (1815-1897), whom he is often quoted as the father of Mathematical Analysis, built the famous real “function Weirstrass ”, which was not differentiable, but nevertheless was con-sidered as a “mathematical curiosity”. However, this curiosity is at the epicenter of the Financial Mathematics. The representations of many exchange and percentage points, are practically of a con-tinuous form as proven by today’s high frequency actual data. But they are of “Boundless variation” throughout the time interval considered. In particular, are indistinguishable (nowhere differentiable), therefore Weierstrass function represents a potential financial graphic. Due to this circumstance, the Financial Calculus, needed an extension for these functions of boundless variation which is task long studied by mathematicians. This gap was discovered along the development of “Stochastic Calculus”, which can be considered as the Theory of Differentiation and Integration of Stochastic Processes. Moreover, reference to “Quadratic Variation”, the “dimensional Itˆo formula”, the “Quadratic Variation Movement” becomes Brownian, the “Stochastic differential of F (x)” and finally, some applications are made.

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