Constrained expected utility maximization: applications in quantitative finance, risk management, life and pension insurance

期望效用最大化在资产定价、最优投资组合理论、金融和精算风险管理中有许多有趣的应用。在真实的情况下，经济主体必须遵循监管限制，最终导致其最优投资决策的重大影响。我的研究项目侧重于风险管理和最优投资组合选择模型在一般情况下的新发展，提供新的灵活性和理论见解，以解决金融和精算科学中的剩余挑战。我的研究计划包括三个长期目标。 我的第一个目标是建立可追踪的风险管理计算框架。例子包括优化经济主体的目标是控制总预期损失的上限的情况，总预期损失被定义为各个预期损失的加权和。我将把结果扩展到在投资时间范围的中间日期或多个投资者的情况下实施中间法规的设置。然后，有趣的是，探讨监管对个人投资行为的纳什均衡的影响。 我的第二个目标，由死亡率风险的存在和参与保险合同的非凹支付结构的动机，集中在一个随机的时间范围内的风险约束下的非凹期望效用问题。我将描述和构造一个算法来数值确定最优策略的死亡时间和金融市场的独立性假设的基础上。研究结果可应用于股票连结寿险合约的设计与风险管理。 由于市场影响和模糊性在最优投资组合选择中起着至关重要的作用，我的第三个目标是研究金融市场中的最优消费和投资问题，其中交易影响未来价格，价格曲线在数量上是非线性的，捕获内生现象，如非线性清算和市场收缩，由于流动性不足。在这样一个非线性内生的永久性市场影响的设置，它是可能的最优性方面的前向-后向随机微分方程（FBSDES）的特征。我计划将模型扩展到一般跳跃模型与时间一致的模糊厌恶的偏好和可能的非凸交易约束。我将通过带跳的FBSDES研究最优解，然后探索保险合同和不可交易衍生品的无差别定价的各种应用。

Expected utility maximization has found many interesting applications in asset pricing, optimal portfolio theory, financial, and actuarial risk management. In real situations, the economic agent has to follow regulatory restrictions that eventually induce significant impacts on her optimal investment decision. My research program focuses on new developments of risk management and optimal portfolio choice models in general settings that give new flexibilities and theoretical insights to solve remaining challenges in finance and actuarial science. My research program consists of three long-term objectives. My first objective is to build tractably computational frameworks of risk management. Examples include the case where the optimizing economic agent aims to control the upper bound of the total expected loss defined as the weighted sum of the individual expected losses. I will extend results to settings with intermediate regulations imposed on prefixed intermediate days of the investment time horizon or with multiple investors. It is then interesting to explore regulatory impacts on the individual investment behavior in a Nash equilibrium. My second objective, motivated by the presence of mortality risk and the non-concave payoff structure of participating insurance contracts, concentrates on a non-concave expected utility problem under a random time horizon with risk constraints. I will characterize and construct an algorithm to numerically determine the optimal strategy based on the independence assumption of the death time and the financial market. The results can find various applications in contract design and risk management of equity-linked life insurance contracts. As market impact and ambiguity play a critical role in optimal portfolio choice, my third objective is to study the optimal consumption and investment problem in a financial market where the trading influences the future prices, and price curves are non-linear in volume, capturing endogenous phenomena as non-linearity in liquidation and market contractions due to illiquidity. In such a non-linear endogenous permanent market impact setting, it is possible to characterize optimality in terms of forward-backward stochastic differential equations (FBSDEs). I plan to extend the model to general jump models with time-consistent ambiguity averse preferences and possibly non-convex trading constraints. I will investigate the optimal solution via FBSDEs with jumps and then explore various applications to indifference pricing of insurance contracts and untradable derivatives.