Research On Some Optimal Investment Problems Under Nonconcave Utilities And Probability Distortions

With the rapid development of the economy and the improvement of the financial system,financial products and derivatives are increasingly rich.Therefore,risk management and portfolio optimization have become important issues.In the research of mathematical finance,investors always make optimal decisions by maximizing their utilities.Behavioral finance theory shows that investors are not completely rational in reality,and their probability perception is subjective,leading to probability distortion.Additionally,due to loss aversion,convex compensation scheme and other factors,the final utility functions may be non-concave.In this paper,we will study some optimal investment problems under non-concave utilities and probability distortions.In Chapter 3,we study the applicability of Lagrange duality in solving the optimal investment problem under general non-concave utilities and probability distortions in depth.By using the concavification and step-wise relaxation,we obtain the optimal solution of the Lagrange unconstrained problem and provide necessary and sufficient conditions on eliminating the Lagrange gap.We also study a special CPT problem and obtain the explicit expression for optimal investment strategy in the Black-Scholes market when the Lagrange strong duality holds.The numerical results show that when the wealth level increases gradually,the proportion of investment in risky assets will tend to a distorted Merton line;the probability distortion function can destroy the time consistency of investment activity.In Chapter 4,we study the optimal investment problem for fund manager with option compensation scheme under rank-dependent utility.By using the splitting method,we divide the original problem into two parts on the benchmark point.We solve the two sub-problems respectively and obtain the optimal splitting point by solving a binary equation.In the Black-Scholes market,we give the analytical solution of optimal investment strategy for the HARA utility manager.Numerical results show that the more incentive fees,the lower the bankruptcy probability and volatility.As the degree of probability distortion function in enlarging the small probability event increases,the fund’s bankruptcy probability and volatility will increase.In Chapter 5,we study the optimal investment problem for manager with traditional compensation scheme under Yaari duality theory and rank-dependent utility theory.Similarly,we solve this problem by using the splitting method and get the optimal investment strategy analytically for the HARA utility manager in Black-Scholes market.Contrary to Yaari’s dual model,numerical results show that under rank-dependent utility,increasing the incentive fee for fund managers or decreasing the initial fundholding will reduce the fund’s loss probability,and decrease the fund’s volatility when the fund runs well.On this basis,Chapter 6 studies the optimal investment problem for manager with traditional compensation scheme under Yaari’s Dual model and VaR/ES risk constraints,and obtain the optimal solution in various cases.The researches of this article have important significance in both theories and applications for solving the general behavioral optimal investment problem with nonconcave utility and guiding investors to make the optimal investment decision.