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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 4, Pages 800–810 (Mi tvp258)

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Nonlinear averaging axioms in financial mathematics and stock price dynamics

V. P. Maslov

M. V. Lomonosov Moscow State University

Abstract: In the presence of an uncertainty factor, that is, if some variable $X$ assumes several values $x_1,\ldots, x_n$ rather than a single value, one usually performs an averaging over these values with some coefficients (measures) $\alpha_i$ such that $\sum_{i=1}^n\alpha_i=1$ and sets $y=\sum\alpha_ix_i$. For an equity market, there arises a nonlinear averaging for $y$. We consider an averaging of the form $f(y)=\sum\alpha_if_i(x_i)$. Starting from four natural axioms, we prove that either the above-mentioned linear averaging holds, or $y=\log\sum_{i=1}^ne^{x_i}$. An example of a stock price breakout under this summation is given.

Keywords: expectation, uncertainty factor, value of a random variable, profit, bank, stock, financial dynamics, stock price breakout.

DOI: https://doi.org/10.4213/tvp258

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English version:
Theory of Probability and its Applications, 2004, 48:4, 723–733

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Citation: V. P. Maslov, “Nonlinear averaging axioms in financial mathematics and stock price dynamics”, Teor. Veroyatnost. i Primenen., 48:4 (2003), 800–810; Theory Probab. Appl., 48:4 (2004), 723–733

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp258
• https://doi.org/10.4213/tvp258
• http://mi.mathnet.ru/eng/tvp/v48/i4/p800

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. P. Maslov, “Zeroth-Order Phase Transitions”, Math. Notes, 76:5 (2004), 697–710
2. V. P. Maslov, “Nonlinear financial averaging, the evolution process, and laws of econophysics”, Theory Probab. Appl., 49:2 (2005), 221–244
3. Maslov V.P., “Quasistable economics and its relationship to the thermodynamics of superfluids. Default as a zero order phase transition”, Russ. J. Math. Phys., 11:4 (2004), 429–455
4. Maslov V.P., “Quasistable economics and its relationship to the thermodynamics of superfluids. Default as a zero order phase transition”, Russ. J. Math. Phys., 11:3 (2004), 308–334
5. Maslov V.P., “An evolution process leading to financial averaging and Gibbs distribution in the limit”, Dokl. Math., 70:1 (2004), 531–534
6. Maslov A.V.P., “Expenditures of purchasers and turnover rate under a nonlinear financial averaging: Laws of economic physics”, Dokl. Math., 69:3 (2004), 348–351
7. Maslov V.P., “The dependence of the purchasing capability and average income of a population on the number of purchasers at specialized markets and in the region. Laws of economic physics”, Dokl. Math., 69:2 (2004), 183–187
8. V. P. Maslov, “Nonlinear Averages in Economics”, Math. Notes, 78:3 (2005), 347–363
9. Maslov V.P., “Quantum economics”, Russ. J. Math. Phys., 12:2 (2005), 219–231
10. Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, 2009, 239–279
11. Kostin V.A., Kostin A.V., Kostin D.V., “Nonlinear Averaging Function of Maslov and Mathematical Models in Economics”, Dokl. Math., 87:2 (2013), 167–170
12. Zegarlinski B., “Linear and Nonlinear Dissipative Dynamics”, Rep. Math. Phys., 77:3 (2016), 377–397
13. Koroliuk D.V. Koroliuk V.S., “A Difference Diffusion Model With Two Equilibrium States”, Cybern. Syst. Anal., 53:6 (2017), 914–924
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