A review is given to the recent developments of the studies for the continuous time Merton portfolio optimization problems. They include risk-sensitive portfolio optimization problems, upside chance and downside risk probabilities optimization and optimal consumption problems. The developments follow by the ideas of Fleming(1995) given in IMA Vol, which suggests to reformulate the risk-sensitive optimization problem as a risk-sensitive stochastic control problem. We will also discuss recent ideas to find the solution for the finite time consumption problem by rewriting the HJB equation as an inf-sup type Isaacs equation, suggested by the idea of market completion.