AI 中文总结
研究重大生活事件引发的风险偏好转变对最优投资的影响，通过结合多种论证方法建立完全非线性抛物方程正凸经典解的存在性，得出最优交易策略由事件前后最优策略连接而成，其凸共轭满足HJB方程。
AI 中文摘要
实证研究表明，重大生活事件会在一段时间内显著增加个人风险厌恶程度。本文聚焦此类由事件引发的风险偏好转变如何影响最优投资。在有限时间范围内，重大生活事件可能独立于金融市场发生，投资者预见风险厌恶可能变化，旨在最大化终端财富的预期效用。我们发现相关的汉密尔顿 - 雅可比 - 贝尔曼（HJB）方程涉及事件后的最优值函数，芬切尔 - 勒让德变换无法使该HJB方程线性化，得到一个带有完全非线性项的抛物方程。通过定点、紧致性和验证论证相结合，我们建立了具有适当增长的完全非线性抛物方程的正凸经典解的存在性。该解的凸共轭满足HJB方程且与事件前的最优值函数一致。最优交易策略通过连接事件前和事件后的最优策略获得，前者用HJB方程的解表示，后者是传统的默顿比率。
英文摘要
Major life events can significantly increase individuals' risk aversion over a sustained period of time, as empirical studies reveal. How such an event-triggered shift of risk preferences impacts optimal investment is the focus of this paper. On a finite time horizon where a major life event may occur independently of the financial market, an investor aims to maximize her expected utility from terminal wealth while foreseeing a potential change in her risk aversion. We find that the associated Hamilton--Jacobi--Bellman (HJB) equation involves the post-event optimal value function (under elevated but fixed risk aversion after the event's occurrence), and the Fenchel--Legendre transform fails to linearize this HJB equation: it yields a parabolic equation with a fully nonlinear term, induced precisely by the post-event optimal value function. Through a combination of fixed-point, compactness, and verification arguments, we establish the existence of a positive convex classical solution with suitable growth to the fully-nonlinear parabolic equation. The convex conjugate of this solution is shown to satisfy the HJB equation and coincides with the pre-event optimal value function. The optimal trading strategy is obtained by concatenating the optimal pre-event and post-event strategies -- the former is expressed in terms of the solution to the HJB equation and the latter is traditional Merton's ratio.
