  We study the optimal portfolio selection problem under relative performance
criteria in the market model with random coefficients from the perspective of
many players game theory. We consider five random coefficients which consist of
three market parameters which are used in the risky asset price modeling and
two preference parameters which are related to risk attitude and impact of
relative performance. We focus on two cases; either all agents have Constant
Absolute Risk Aversion (CARA) risk preferences or all agents have Constant
Relative Risk Aversion (CRRA) risk preferences for their investment
optimization problem. For each case, we show that the forward Nash equilibrium
and the mean field equilibrium exist for the n-agent game and the corresponding
mean field stochastic optimal control problem, respectively. To extend the
n-agent game to the continuum of players game, we introduce a measure dependent
forward relative performance process and apply an optimization over controlled
dynamics of McKean-Vlasov type. We conclude that our optimal portfolio formulas
extend the corresponding results of the market model with constant
coefficients.
