Optimal Investment Under Safety First Expected Utility With Distortion

Optimal investment problem is one of the most important parts in the field of financial mathematics.The safety first principle and the expected utility theory are extremely influential theories in the portfolio field,but they are based on different investment objectives,therefore they are both one-sided.By combining the objective functions of the two models with weights,Levy developed a new theoretical model,namely the safety first expected utility model(SFEU model).Such model fills up some of the defects of the traditional safety first principle,but for the phenomena in reality that cannot be explained by the classical expected utility theory,such as Allais Paradox,the SFEU model is also not convincing.In order to make up for the defects of the model,this paper introduces a probability distortion(weighting)function according to the rank dependence theory(RDEU),which distorts the probability of the real world,so that the model can better explain the behavior of investors in the real world.In this paper,we study a portfolio selection problem under safety first expected utility model with distortion(SFEUD model)where the underlying probability scale is transformed by a nonlinear distortion function.By employing the quantile formulation and the relaxation method,we obtain the general form of optimal solution to the problem,and the necessary and sufficient condition for existing such an optimal solution is also proved.What's more,we carried out numerical analysis in a special case.We further demonstrate an analytical solution to the optimal strategies under four different monotonic cases.What's more,the optimal terminal wealth process is replicated into a portfolio of options and deposits under specific circumstances.