Suppose that we are given a semi-martngale whose coefficients are affected by a diffusion process and a control parameter. We are concerned with minimizing the probability that the growth rate of the semi-martingale falls below a certain level over large time. It is to be noted that the asymptotic behavior of the minimizing probability relates to risk-sensitive stochastic control in the risk-averse case. Thus, by anlyzing the correponding HJB equation to the risk-sensitive control problem, we shall give the exponential decay rate of the minimizing probability and an (asymptotically) optimal strategy attaining the rate. The problem is motivated by Mathematical Finance and we shall also mention applications to optimal investment models.