We consider the probability of falling below a target growth rate of the wealth process, or its ratio to a preset portfolio taken as a benchmark, up to time horizon T.
Then we study the asymptotic behavior of minimizing probability as T tends to infinity. We formulate this problem as "large deviation control", in which we are concerned with asymptotic behavior of the minimizing probability regarding as an extension of the large deviation principle. More precisely, we aim to find the rate function of the asymptotics, its "effective domain" where the rate function does not vanish nor diverge, and then asymptotically optimal strategies which realize the asymptotics. We also discuss the problem under model uncertainty.|