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This paper studies the problem of optimal investment with CRRA (Constant, Relative Risk Aversion) preferences, subject to dynamic risk constraints on trading strategies. The market model considered is continuous in time and incomplete; furthermore, financial assets are modeled by It o processes. The dynamic risk constraints (time, state dependent) are generated by risk measures. The optimal trading strategy is characterized by a quadratic BSDE. Special risk measures (Value-at-Risk, Tail Value-at-Risk and Limited Expected Loss) are considered and a three-fund separation result is established in these cases. Numerical results emphasize the effect of imposing risk constraints on trading.
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