Studies On Dual Control Methods For Financial Insurance Problems Under Stochastic Volatility Models

The optimal allocation of assets is primarily affected by factors such as the stock price process and investor risk preferences in finance and insurance.Therefore,it is particularly important to accurately portray stock prices and investor risk appetite.With the assumption that stock prices obey stochastic volatility models such as the dynamic elasticity of variance（DEV）model,the multi-factor Heston model,and the rough Heston model,investors have general utility functions such as symmetric asymptotic hyperbolic absolute risk aversion（SAHARA）utility and S-shaped utility.This thesis investigates the optimal investment problem for defined contribution（DC）pension,the optimal reinsurance investment problem,and the optimal reinsurance investment game problem.Although the models considered in this thesis can describe the stock price process and investor risk preference more closely,it is impossible to obtain explicit solutions for the optimal strategies.In this thesis,the dual control methods are developed to solve the approximate solutions of the optimal strategies,and sensitivity analysis is carried out.The main works are as follows.Firstly,we investigate the optimal investment problem of the DC pension plan in incomplete financial markets.Assume that the financial market consists of a risk-free asset,a risky asset whose price process obeys the DEV model,and a correlated inflation risk model.Under the SAHARA utility,the pension participants aim to maximize the expected utility of terminal wealth and construct a utility maximization problem with Markov characteristics and concave properties.We derive the Hamilton-Jacobi-Bellman equations and the verification results,develop lower and upper bounds for the value function and compute the approximation solutions of optimal strategies by applying the dual control Monte Carlo algorithm.Secondly,the optimal reinsurance investment problem is considered.Assume that the insurer can purchase proportional reinsurance and invest in a financial market composed of a risk-free asset and a risky asset whose price process obeys the multi-factor Heston model.Under the S-shaped utility,the insurer aims to maximize the expected utility of the terminal wealth with a minimum guaranteed threshold and construct a utility maximization problem with Markov characteristics but not concave properties.We transform the non-concave utility maximization problem by applying coalification to solve a concave utility maximization problem.Finally,we derive the HJB equation and apply the dual control methods to calculate the approximate solutions of the optimal strategies and the value functions.Then,we study the optimal reinsurance investment problem again,but assume that the stock process obeys a rough Heston model with non-Markovian and non-semimartingale characteristics.We construct a utility maximization problem with neither Markovian nor concave properties and transform it into a utility maximization problem with Markovian properties and concave through concave closure and semi-martingale approximation.The convergence analysis of the optimal strategies and the proof of the convergence order is also given.The approximate problem is a kind of classical stochastic control problems.Thus we derive the HJB equation and apply the dual control methods to calculate the approximate solutions of the optimal strategies and the value functions.Finally,the optimal reinsurance investment game between two insurers is discussed.Assume that the insurer can purchase proportional reinsurance contracts and invest in a financial market composed of a risk-free asset and a risky asset whose price process obeys a constant elasticity of variance（CEV）model.In addition,assume that the insurance market risk is related to the stock market risk.Under the SAHARA utility,we construct the utility maximization problem to maximize the relative terminal wealth utility.We derive the HJB equation and obtain the approximate solutions of the value functions and Nash equilibrium strategies by constructing a strong duality.