AI 中文总结
本文构建多活动下里程碑驱动的初创企业随机控制模型，求解出仅依赖有效前沿曲线的最优策略，为创业决策提供了基础支撑。
AI 中文摘要
新创企业需通过多个阶段达成里程碑目标以实现“生存”。本文研究里程碑导向场景下初创企业的策略，构建了一个创业初创企业模型，其状态由扩散过程描述，创业者可在多项活动（或控制项）间进行选择，这些活动具有不同成本，并决定该过程的漂移项与方差。根据过程达到固定上边界还是下边界，初创企业将取得成功或失败。具有多项（≥3个）控制项的连续时间随机模型通常极难处理，本文成功完全求解了最优策略，并对其结构给出明确刻画。具体而言，最优策略仅使用由所谓有效前沿曲线所表征的一组控制项，该曲线通过两个直观指标对控制项排序：风险性（漂移与波动率之比）和成本效益（漂移与成本之比）。本文模型的独特之处在于，根据模型参数的不同，有效前沿曲线可呈现不同类型，进而导致最优策略在结构上存在质的差异。据所知，这是首个分析允许不同类型有效前沿曲线的随机控制模型的研究，为初创企业提供了评估其活动的直观指标，并为里程碑导向场景下的最优策略如何随具体场景发生质性变化提供了有价值的见解，研究结果为创业决策研究提供了基础支撑。
英文摘要
New venture start-ups need to ``survive'' through multiple stages of reaching milestone targets. We investigate the strategies for start-ups in a milestone-oriented setting. We examine a model of an entrepreneurial start-up firm, where its state is captured by a diffusion process. The entrepreneur can choose between multiple activities (or controls), which incur different cost and determine the drift and the variance of the process. Depending on whether the process reaches a fixed upper boundary or a lower one, the start-up firm succeeds or fails. Continuous-time stochastic models with multiple ($\ge 3$) controls are typically very challenging to deal with. In this work, we are able to completely solve for the optimal policy and provide an explicit characterization of its structure. In particular, the optimal policy only uses controls from a set characterized by a so-called efficient frontier curve that orders the controls by two intuitive measures: riskiness (drift-to-volatility ratio) and cost-effectiveness (drift-to-cost ratio). A unique feature of our model is that depending on the model parameters, the efficient frontier curves can be of different types, resulting in qualitatively different structures of the optimal policy. As far as we know, this is the first study that analyzes a stochastic control model which admits efficient frontier curves of different types. Our work provides start-up firms with intuitive measures to evaluate their activities and offers valuable insights on how the optimal strategies in a milestone-oriented setting change qualitatively contingent upon the specific scenario. We believe the results provide a foundational block in the study of entrepreneurial decision-making.
