I have a set of nonlinear equations to solve which came up in my research.
I take the conditional expected value of $N$ functions (which are $log()$) of $N$ independent non-identically distributed exponential random variables $X_1,..,X_i,..,X_j,...X_N$. These random variables are all known by their mean values.
Then I obtain the set of functions $ f_1(\cdot),..,f_i(\cdot),..., f_N(\cdot)$, where the $i^{th}$ function $f_i(\cdot)$ is as follows:
$$f_i(d_1,...,d_N)=[P(A_i)\cdot E_{X \mid A_i}\log(1+\cfrac{d_i}{\sum_{j \neq i} X_j})] -c_i$$
The summation $\sum_{j \neq i} X_j$ is the sum of all the $N-1$ random variables except $X_i$ i.e., sum over all $j=\{1,...,N\}\smallsetminus\{i\}$
The probability of the event $A_i$ is given by $P(A_i)=\prod_{j\neq i} P(X_j>d_j, X_j=\max\{X_j,Y_j,Z_j\})$. Here too the product is over all $j=\{1,...,N\}\smallsetminus\{i\}$
For a given $j={1,...,N}$ the three random variables $X_j, Y_j, Z_j$ are i.i.d., i.e., the $N$ sets $\{X_j, Y_j, Z_j\}$ for $j=1,...N$ are individually i.i.d. The $c_i$ for $i=1,...N$ are positive constants.
Then I have $N$ functions of the $N$ unknowns $d_i$ for $i={i,..., N}$.
Then I want to find a root of this set of functions such that $f_i=0$ for all $i$.
Is there a method to solve it or is there some criteria that would assure the existence of a solution?
I have tried Banach's fixed point theorem. But find it difficult to find a metric so that this set of functions has a contraction.
Thank you.
It seems that the setting is the following. One considers some positive real numbers $c_i$ and some independent random variables $X_i$, $Y_i$, $Z_i$, all exponential, where the common parameter of $X_i$, $Y_i$ and $Z_i$ is some positive $\lambda_i$. Let $\lambda=(\lambda_i)_i$ and $S=X_1+X_2+\cdots+X_N$. For every $x=(x_i)_i$ with positive entries, and every $i$, let $\hat x_i=(x_j)_{j\ne i}$ and $$ A_i(\hat x_i)=[\forall j\ne i, X_j\geqslant\max\{Y_j,Z_j,x_j\}]. $$ The question is to find some $x=(x_i)_i$ with positive entries such that $f_i(\lambda,x)=c_i$ for every $1\leqslant i\leqslant N$, where $$ f_i(\lambda,x)=E\left[\log\left(1+\frac{x_i}{S-X_i}\right);A_i(\hat x_i)\right]. $$ To solve this system explicitely seems hopeless in the general case. To show that some solution exists (or not), one could study the functions $f_i(\lambda,\ )$ on the boundary of their domain. And maybe study in depth the case $N=2$...