$X_t=\phi X_{t-1}+W_t$
$\Rightarrow X_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$
Where , $|\phi|>1$ .
But how does the following recursion relation occur : $$X_t=-\sum_{j=1}^{\infty}\phi^{-j}W_{t+j}$$ ?
I tried to do this by Again writing :
$X_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$
$\Rightarrow\phi X_{t-2}+W_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$
$\Rightarrow\phi [\frac{1}{\phi} X_{t-1}-\frac{1}{\phi}W_{t-1}]+W_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$
But it seems i have gone to the same equation .
As you stated, $X_t=\phi X_{t-1}+W_t$, suggesting $X_t=\frac1\phi(X_{t+1}-W_{t+1})=-\phi^{-1}W_{t+1}+\phi^{-1} X_{t+1}$. Substituting this recursively, it follows that $$\begin{align*}X_t&=-\phi^{-1}W_{t+1}+\phi^{-1}(-\phi^{-1}W_{t+2}+\phi^{-1}X_{t+2})\\&=-\phi^{-1}W_{t+1}-\phi^{-2}W_{t+2}+\phi^{-2}X_{t+2}\\&=\dots\\&=-\phi^{-1}W_{t+1}-\phi^{-2}W_{t+2}-\phi^{-3}W_{t+3}+\dots\\&=-\sum_{j=1}^\infty \phi^{-j}W_{t+j}\end{align*}$$i.e. as an expansion in terms of $\phi^{-1}$ with coefficients $-W_{t+j}$. This converges if $W_{t+j}$ grows slow enough, i.e. $W_{t+j}\in O(r^j)$ such that $|r/\phi|<1$.
Is this meant as a stochastic difference equation? $W_t$ a Wiener process?