Ergodic problems for linear exponential quadratic Gaussian control and linear quadratic stochastic differential games

In this paper a direct approach is given for the verification of the relation between the Riccati equation and the optimal cost for the solution of a linear exponential quadratic Gaussian control problem and the Riccati equation and the optimal payoff for the solution of a linear quadratic stochastic differential game. The cost functionals are expected long run averages. It is shown in a direct way why the two solutions are naturally related and why the ergodic costs are the same. While the equality of the two ergodic costs is known, the approach given here should provide further insight into this equality and the relation of these two solutions.