Solving the Problem of Choosing the Optimal Risk Sharing in a Static Problem of Insurance with a Limit on the Mean Risk of the Insurer

The object of this research is the process of risk sharing between participants in an insurance transaction. More accurate, we need to find the optimal risk sharing function between the insurer and the client in the basic insurance model. To that end, an optimization problem will be formulated. It is necessary to find the maximum value of the target functional that characterizes the insurer's utility when limiting the average risk of the client (the average risk of the client should not exceed a certain value). First, we have to find the general solution of this problem. Then, we will deal with a numerical example for a specific type of damage distribution. In the course of the work, we will apply information from probability theory and optimization methods, in particular, the Kuhn – Tucker theorem and the Neyman– Pearson Lemma. In addition, the Python programming language will be used to obtain numerical solutions and plot graphs.