Optimal portfolio selection in stochastic environment

In this dissertation, we consider a particular case of an optimal consumption and portfolio selection problem for an innitely lived investor whose consumption rate process is subject to downside constraint. We also suppose that the wealth dynamics is composed of three assets (i) riskless assets (ii) risky assets (iii) hedge assets. We consider the investor''s wealth process, interpreted in the sense of the It^o integral.Our work aims to find the optimal policies which maximize the expected discount utility function.Furthermore, we obtain the optimal policies in an explicit form for the log utility function which is a special case of the general utility (CRRA) function, using the martingale method and applying the Legendre transform formula and the Feynman-kac formula. We derive some numerical results for the optimal policies and compare the results with the classical Merton''s result evaluated for an innite horizon case.