Optimal Monotone Mean-variance Problem And Controlled Jump Diffusion Processes In Insurance Risk Theory

In this thesis,we first consider the optimal reinsurance and investment problem for insurance company using monotone mean-variance utility as the objective of the optimization problem.In Chapters 2-4,we use a diffusion approximation model,a Cram(?)r-Lundberg model,and a catastrophe insurance model simulated by a Cox process driven by a shot-noise process to model the surplus process of an insurance company,respectively.For each of these models,an explicit optimal value function and an explicit optimal strategy are obtained,respectively.We also obtain the efficient frontiers corresponding to these models.After that,by using the Gamma process to model the surplus process of the insurance company,we consider the problem of maximizing the expected utility of the terminal wealth and minimizing the ruin probability under this process.For this former problem,we also obtain an explicit optimal value function and an explicit optimal strategy;for the latter,the expressions for the optimal value function and the optimal strategy are obtained,which can be expressed via the roots of a pair of elementary equations,and we subsequently prove the existence of the roots of this pair of elementary equations and give numerical solutions.This thesis is organized as follows.Chapter 1 is the preliminary,which introduces the definition of monotone mean-variance utility and some mathematical methods.Chapter 2 studies the optimal monotone mean-variance problem under the diffusion approximation model,and obtains the efficient frontier and monotone CAPM.Chapter 3 studies the optimal monotone mean-variance problem under the Cram(?)r-Lundberg model,and obtains the efficient frontier and presents numerical examples.Chapter 4 first introduces the catastrophe insurance model simulated by a Cox process driven by a shot-noise process,and then investigates the optimal monotone mean-variance problem under this model,and also obtains the efficient frontier.Chapter 5 investigates the optimization problem for the Gamma process.