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Nonlinear averaging axioms in financial mathematics
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 800-810 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, bank, stock, financial dynamics, stock price breakout.
Mots-clés : profit 
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V. P. Maslov. Nonlinear averaging axioms in financial mathematics. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 800-810. http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a9/

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