We shall discuss on large deviation estimates of the probability of falling below a given target growth rate for controlled semimartingales, in relation to certain ergodic risk-sensitive stochastic control problems in the risk averse case. We present an expression of the limit value of the minimizing probability as the Legendre transform of the value of the stochastic control problem on infinite time horizon, which is characterized as the solution to the H-J-B equation of ergodic type. The problems discussed here are motivated by mathematical finance and they are called “downside risk minimization”. In this chapter, after reviewing the probabilistic meaning of the asymptotic analyses developed here and historical situation of the studies, we make an exposition about how we can obtain our duality theorem on the asymptotics.