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。在现实情况下，经济代理人必须遵守监管限制，这些限制最终会对她的最优投资决策产生重大影响。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环境下的新发展，这些模型提供了新的灵活性和理论见解，以解决金融和精算科学中剩余的挑战。我的研究计划包括三个长期目标。我的第一个目标是建立易于计算的风险管理框架。例如，最优经济主体的目标是控制总预期损失的上界，该上限被定义为单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前置中间天实施中间法规或与多个投资者一起实施中间法规的设置。那么，在纳什均衡中探索监管对个人投资行为的影响就很有趣了。我的第二个目标是在死亡风险的存在和参与保险合同的非凹收益结构的激励下，集中在具有风险约束的随机时间范围下的非凹期望效用问题。基于死亡时间和金融市场的独立性假设，刻画并构造了一个数值确定最优策略的算法。研究结果可在股权连结寿险合约设计和风险管理中得到广泛应用。由于市场影响和模糊性在最优投资组合选择中起着关键作用，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，捕捉到清算过程中的非线性和流动性不足导致的市场收缩等内生现象。在这种非线性的内生永久市场影响的背景下，可以用正倒向随机微分方程组来刻画最优性。我计划将该模型扩展到具有时间一致性、模糊性、厌恶偏好和可能的非凸交易约束的一般跳跃模型。我将通过带跳跃的FBSDE来研究最优解，然后探索在保险合同和不可交易衍生品的无差别定价中的各种应用。

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.