  In this paper, we analyze linear-quadratic stochastic differential games with
a continuum of players interacting through graphon aggregates, each state being
subject to idiosyncratic Brownian shocks. The major technical issue is the
joint measurability of the player state trajectories with respect to samples
and player labels, which is required to compute for example costs involving the
graphon aggregate. To resolve this issue we set the game in a Fubini extension
of a product probability space. We provide conditions under which the graphon
aggregates are deterministic and the linear state equation is uniquely solvable
for all players in the continuum. The Pontryagin maximum principle yields
equilibrium conditions for the graphon game in the form of a forward-backward
stochastic differential equation, for which we establish existence and
uniqueness. We then study how graphon games approximate games with finitely
many players over graphs with random weights. We illustrate some of the results
with a numerical example.
