Break equation in parts
After removing all unnecessary brackets, correcting size of used ones, with use nccmath
for medium size fractions and changed lines of equation:
\documentclass{article}
\usepackage{nccmath}
%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
\begin{document}
\begin{multline*}
P_t=
\frac{1}{2}\Biggl(m_t+v_t-u_t \\
+ \frac{1-\zeta }{2}\frac{E_{t-1}m_t
+ \mfrac{1-\zeta}{2\zeta}E_{t-2}m_t
+ \mfrac{\zeta(1-\zeta)}{2(1+\zeta)}E_{t-2}P_{t-1}
+ \mfrac{\zeta}{2}P_{t-1}
+ \mfrac{\zeta}{1+\zeta}P_{t-2}}
{1+\zeta}\Biggr) \\
+ \frac{1-\zeta}{4}
\frac{E_{t-2}m_t + \frac{\zeta}{2}(E_{t-2}P_{t-1}
+ P_{t-2})}{1+\zeta}
+ \frac{\zeta }{4}\Bigl(P_{t-1}+P_{t-2}\Bigr)
\end{multline*}
\end{document}
(red lines indicate text borders)
I would simplify the formula first and then typeset it across three rows using an align*
environment.
\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\E}{E} % expectations operator
\begin{document}
\begin{align*}
P_t
&=\frac{1}{2}(m_t+v_t-u_t)
+\frac{1}{4}\frac{1-\zeta}{1+\zeta}
\biggl[\E_{t-1}m_t +\frac{1-\zeta}{2\zeta }\E_{t-2}m_t\\
&\quad +\frac{\zeta (1-\zeta)}{2(1+\zeta)}\E_{t-2}P_{t-1}
+\frac{\zeta}{2}P_{t-1}+\frac{\zeta}{1+\zeta }P_{t-2}
+\E_{t-2}m_t\\
&\quad +\frac{\zeta}{2} (\E_{t-2}P_{t-1}+P_{t-2})\biggr]
+\frac{1}{4}\zeta (P_{t-1}+P_{t-2})
\end{align*}
\end{document}