continuous time portfolio optimization

this paper presents dynamic portfolio model based on the merton's optimal investment-consumption model, which combines dynamic synthetic put option using risk-free and risky assets. this paper is extended version of methodological paper published by yuan yao (2012) cite{26}. because of the long history of the development of foreign financial market, with a variety of financial derivatives, the study on theory or empirical analysis of portfolio insurance focused on how best can portfolio strategies be used in minimizing risk and market volatility. in this paper, stock and risk-free assets are used to replicate options and to establish a new dynamic model to analyze the implementation of the dynamic  process of investors' actions using dynamic replication strategy. our results show that investors' optimal strategies of portfolio are not dependent on their wealth, but are dependent on market risk and  this new methodology is broaden in compare to paper of yuan yao (2012).