发表机构
University of Warwick; Shanghai University of Finance and Economics; Southern University of Science and Technology（华威大学; 上海财经大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对投资者在达到财务目标或外部截止日期前的最优投资问题，提出随机控制框架，以贴现函数衡量目标达成时间满意度，建立贝尔曼最优性原理并将价值函数刻画为HJB方程粘性解，开发霍华德算法求解数值解，发现最优策略可随风险资产漂移递减且不收敛于无风险投资。
AI 中文摘要
我们为一个投资者开发了一个框架，该投资者进行交易直到她达到财务目标或外部截止日期到来。类似于财富上的效用函数，我们通过贴现函数来衡量达到目标时机的满意度。对于连续时间市场，其中随机因子驱动股票价格和财务目标的动态，投资者最大化在目标达到时的期望贴现，以及如果目标在截止日期前未达到，则最大化资金比率的期望效用。这种设置导致了一类新的随机控制问题。我们建立了贝尔曼最优性原理，并将价值函数刻画为相关哈密顿-雅可比-贝尔曼方程的粘性解。当截止日期无限且目标恒定，HJB方程简化为与倒向热方程相关的形式，我们为其提供了光滑解的完整刻画。当解析解不可用时，我们开发了一种基于霍华德算法的方法来获得数值解。我们的分析表明，目标达到问题的最优投资策略可以随风险资产的漂移而递减，并且即使波动率发散到无穷大，也不一定收敛到完全无风险投资。
英文摘要
We develop a framework for an investor who trades until she either reaches a financial goal or an exogenous deadline arrives. Analogous to utility functions over wealth, we measure satisfaction with the timing of reaching a goal by a discount function. For a continuous-time market where a stochastic factor drives the dynamics of stock prices and the financial goals, the investor maximizes the expected discount at the goal reaching time and the expected utility of the funding ratio if the goal remains unreached by the deadline. This setup leads to a new class of stochastic control problems. We establish Bellman's principle of optimality and characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman equation. When the deadline is infinite and the goal is constant, the HJB equation reduces to a form related to the backward heat equation, for which we provide a complete characterization of smooth solutions. When analytical solutions are unavailable, we develop an approach based on Howard's algorithm to obtain numerical solutions. Our analysis shows that optimal investment policies for goal reaching problems can be decreasing in the drift of risky assets and need not converge to full risk-free investment even as volatility diverges to infinity.
