  We introduce and solve a new type of quadratic backward stochastic
differential equation systems defined in an infinite time horizon, called
\emph{ergodic BSDE systems}. Such systems arise naturally as candidate
solutions to characterize forward performance processes and their associated
optimal trading strategies in a regime switching market. In addition, we
develop a connection between the solution of the ergodic BSDE system and the
long-term growth rate of classical utility maximization problems, and use the
ergodic BSDE system to study the large time behavior of PDE systems with
quadratic growth Hamiltonians.
