New Venture Creation: A Drift-Variance Diffusion Control Model

Abstract

We model the creation of a new venture with a novel drift-variance diffusion control framework in which the state of the venture is captured by a diffusion process. The entrepreneur creating the venture chooses costly controls, which determine both the drift and the variance of the process. When the process reaches an upper boundary, the venture succeeds and the entrepreneur receives a reward. When the process reaches a lower boundary, the venture fails. The entrepreneur can choose between two different controls and wishes to determine the policy that maximizes the expected total reward minus total cost. We consider two variations of the model: one in which both boundaries are fixed and one in which only the upper boundary is fixed but the lower is free. We derive closed-form expressions under which the optimal policy will be dynamic versus static; we prove that when the policy is dynamic, it switches between the two controls at most once. The results reveal a subtle trade-off between the cost of the two controls, their drift and their variances, in which controls that are more expensive may be utilized more than controls that are less expensive. We also demonstrate that in the fixed boundary case, the entrepreneur may wastefully use a more expensive control near the lower boundary to avoid hitting that boundary. This implies that efficient utilization of the two controls cannot happen when the entrepreneur does not have the freedom to choose when to abandon the venture.