Optimal Control Problem For Multi-Scale Reinsurance Models With Stochastic Volatilities

In this thesis,we mainly study the limit behavior for a class of optimal control problems and solutions when multi-scale reinsurance models with stochastic volatilities evolve on a fast time scale.The reinsurance optimal investment strategy problem is a popular and essential problem in financial mathematics,especially the singular perturbation problem with small parameters,which has attracted widespread attention of scholars.It is of great significance to consider the limit behavior of the solutions and the reduced models because of the complexity of solving multi-scale system problems and the difficulty to calculate the singular perturbed problems.It not only simplifies the complexity of the model,but also reduces the computational difficulty in solving the optimal problem.Firstly,we apply the effective Hamiltonian method to obtain the limit behavior of the solution of the optimal control problem when the small parameter ε tends to zero.The only viscosity solution of the HJB equation determined by the dynamic programming principle uniformly converges to the unique viscosity solution of the HJB equation in the effective control system composed of the effective Hamiltonian and the effective terminal condition on the compact set.We also obtain the reduced model of the original problem and the explicit expression of the effective Hamiltonian for easy calculation.Secondly,we employ the perturbation analysis method to give the uniformly effective secondorder approximate solution of the viscosity solution of the HJB equation of the reinsurance financial model and establish the error correction model of the original model,and the accuracy is improved.Finally,we consider the stochastic differential game models and the optimal control problems of several insurance companies.We obtain the convergence results of the solution of the optimal control problems and the explicit expressions of the effective HJBI operators through the effective Hamiltonian and iterative method,we also establish the effective macro-reduction model of the original financial model.