Nonlinear averaging axioms in financial mathematics and stock price dynamics

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.