  We study the convergence problem for mean field games with common noise and
controlled volatility. We adopt the strategy recently put forth by Lauri\`ere
and the second author, using the maximum principle to recast the convergence
problem as a question of ``forward-backward propagation of chaos", i.e
(conditional) propagation of chaos for systems of particles evolving forward
and backward in time. Our main results show that displacement monotonicity can
be used to obtain this propagation of chaos, which leads to quantitative
convergence results for open-loop Nash equilibria for a class of mean field
games. Our results seem to be the first (quantitative or qualitative) which
apply to games in which the common noise is controlled. The proofs are
relatively simple, and rely on a well-known technique for proving
well-posedness of FBSDEs which is combined with displacement monotonicity in a
novel way. To demonstrate the flexibility of the approach, we also use the same
arguments to obtain convergence results for a class of infinite horizon
discounted mean field games.
