# Solving a non-linear inequality related to geometric Brownian motion

Consider the non linear inequality $$\sum_{i=1}^{n}a_{i}u^{\sum\limits_{j=1}^{i}y_j} > c$$ $$y_j \in \{0,1\}, j=1,2,\dots,n$$ $$a_i \in \mathbb{R}, i=1,2,\dots,n$$ $$n \in \mathbb{N}, u>0, c > 0$$ where the $y_j$ are variables and all other term constant. Is there a way to find the solution space analytically?

The inequality arises as follow: start with a geometric Brownian motion process $(S_t)_{t \geq 0}$, approximate it by a binomial process, define the random variable $X=\sum\limits_{i=1}^{n}a_iS_n/S_{i-1},n \in N$, evaluate $E(\max(X,c)), c>0$.   The inequality is an attempt to calculate how many terms are involved in the limited expectation

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I think the question is rather related to analysis issues then to probability ones, so I've added the tag [inequality]. Do you have any information on $u$ if it is smaller or bigger then $1$? – Ilya Oct 29 '11 at 11:45
@Gortaur Thanks for adding, I wasn't sure on how to tag the question. Depending on how I parametrize my binomial approximation of the geometric brownian motion, $u \in (0,1)$ or $u \in (1,2)$. – Nicolas Essis-Breton Oct 29 '11 at 15:30