This survey reviews portfolio selection problem for long-term horizon. We
consider two objectives: (i) maximize the probability for outperforming a
target growth rate of wealth process (ii) minimize the probability of falling
below a target growth rate. We study the asymptotic behavior of these criteria
formulated as large deviations control pro\-blems, that we solve by duality
method leading to ergodic risk-sensitive portfolio optimization problems.
Special emphasis is placed on linear factor models where explicit solutions are
obtained
