Optimal Strategies for Milestone-Driven Startups in Multi-Activity Settings

Start-up ventures operate in a dynamic environment, constantly striving to achieve specific milestones to ensure their survival and growth. This study delves into the optimal strategies for such milestone-oriented start-ups, particularly when they can choose from multiple activities, each with varying costs and impacts on their progress. The research models the start-up's state as a diffusion process, where the entrepreneur's choice of activity influences both the drift (progress) and variance (risk) of this process. The ultimate goal is to reach an upper boundary (success) while avoiding a lower boundary (failure). Dealing with continuous-time stochastic models involving three or more controls is typically complex, but this work successfully provides a complete solution for the optimal policy. A key finding is that the optimal policy exclusively utilizes activities from a set defined by an 'efficient frontier curve.' This curve orders activities based on two intuitive metrics: riskiness (the ratio of drift to volatility) and cost-effectiveness (the ratio of drift to cost). Uniquely, the model reveals that the structure of the optimal policy can qualitatively change depending on the type of efficient frontier curve, offering valuable insights into entrepreneurial decision-making under different scenarios.