This paper shows that the long-time behavior of the entropic risk measure (under both forward performance process framework and classical utility framework) converges to a constant, which is independent of the initial state of the stochastic factors in a stochastic factor model. The exponential convergence rate to the long-term limit is also obtained by using ergodic backward stochastic differential equation method. Finally, the paper establishes a connection between the two notions of entropic risk measures and their large time behavior.
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