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Optimal investment problem under behavioral setting: A Lagrange duality perspective

**Speaker**: Zhenyu Cui from Stevens Institute of Technology

**Time**: 02/18/2021, 2:00pm -2:50pm
- Zoom ID: 992 7853 8762
**Abstract**:We consider the optimal investment problem with both probability distortion/weighting (e.g. inverse S-shaped probability weighting) and general non-concave utility functions (e.g. S-shape utility). The regular method to solve this type of problems is to apply the concavification technique and then refer to Lagrange duality to generate the optimal solutions. Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the appearance of probability weighting or the non-concavity of utility functions, but not both. In contrast, we have shown that this strong duality relationship would not be maintained unconditionally if we try to combine both factors together. A step-wise relaxation is proposed to handle miscellaneous general non-concave utility and probability distortion functions. The necessary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to a couple of typical examples in behavioral finance including the CPT model, VAR-RM model with distortions, Yarri’s dual model and the goal reaching model. The closed form solutions on the optimal trading strategies are obtained for a special example of the CPT model where a ”distorted” Merton line has been shown exactly.