Fix $N\in\mathbb{N}$. For any $i\in\{1,...,N\}$, \begin{equation} X^i_t=x_i\exp\left(\sigma B_t^i\right) \end{equation} where $B^i$ are independent standard Brownian motions, and $x_i>0$.
Question:Let $r_i(x)=\frac{x_i}{\sum_{k=1}^Nx_k}$ for any $x\in\mathbb{R}_+^N$, then what is $\mathbb{E}[r_i(X_T)]$?
My attempt is the martingale approach as follows:
Let $f(t,x)=\mathbb{E}[r_i(X_T)|X_t=x]$, then apply Ito lemma and set the drift to zero since $f(t,X_t)$ is a martingale we obtain the following PDE
\begin{equation} f_t+\frac{\sigma^2}{2}\sum_{k=1}^Nx_kf_{x_k}+\frac{\sigma^2}{2}\sum_{k=1}^Nx_k^2f_{x_kx_k}=0 \end{equation} with the terminal condition $f(T,x)=r_i(x)$ for any $x\in\mathbb{R}_+^N$. But solving this PDE is troublesome for me.