Before revision:  We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical power robust expected utility for long time horizons  .
After revision:  We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical power robust expected utility for long time horizons . Finally, we give an example to illustrate that our approach can be applied to address a type of robust forward investment performance processes with negative realization processes .