  In this paper, we propose a mean-field game model for the price formation of
a commodity whose production is subjected to random fluctuations. The model
generalizes existing deterministic price formation models. Agents seek to
minimize their average cost by choosing their trading rates with a price that
is characterized by a balance between supply and demand. The supply and the
price processes are assumed to follow stochastic differential equations. Here,
we show that, for linear dynamics and quadratic costs, the optimal trading
rates are determined in feedback form. Hence, the price arises as the solution
to a stochastic differential equation, whose coefficients depend on the
solution of a system of ordinary differential equations.
