Portfolio Optimization under Habit Formation
  The "standard" Merton formulation of optimal investment and consumption involves optimizing the integrated lifetime utility of consumption, suitably discounted, together with the discounted future bequest. In this formulation the utility of consumption at any given time depends only on the amount consumed at that time. However, it is both theoretically and empirically reasonable that an individuals utility of consumption would depend on past consumption history. Economists term this "Habit Formation". We introduce a new formulation of habit formation which allows non-addictive consumption patterns for a wide variety of utility specification. In this paper we construct a simple mathematical description of this habit formation and present numerical solutions. We compare the results with the standard ones and draw insights obtained from the habit formation. The consumption path tends to increase with time and be less sensitive to the market fluctuations, which perfectly reflects the existence of habit persistence of an investor. At the same time, his decreasing risk aversion, which seems to be in contradiction with the empirical evidence, can be explained within the limitations of the model. 
Roman Naryshkin and Matt Davison