Essays in stochastic modeling with applications in entrepreneurship and healthcare

In this thesis we present applications of stochastic modeling in entrepreneurship and healthcare. In Chapter 1 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 and 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. In Chapter 2 we model entrepreneurs' behavior when they engage in a simultaneous process of search and hypothesis testing. Entrepreneurs search for a set of strategic and operational choices that will maximize their venture's profits and they test the hypothesis that these profits exceed a minimum threshold of viability. We formulate a problem of the entrepreneur opportunity search process, where in each time period, the entrepreneur can stop and conclude, or choose an experiment from a set of strategic and operational options, implement it and observe the resulting profit. Using tools from machine learning to model the search process and tools from sequential hypothesis testing to model the testing procedure, we analytically characterize the optimal testing strategy for the resulting problem. We demonstrate that in certain scenarios the optimal testing strategy from our framework and that predicted by the Lean Startup Theory are consistent, while in others they disagree. In Chapter 3 we present an application of stochastic modeling in healthcare. Conventional clinical trial designs adopt a fixed sample size and assume a standard type I error of 5\% and type II error of 10\% or 20\%. While interim review of the results is possible to accelerate decision making, these designs are rigid because they do not allow changes in the design over time in response to the results accumulated, or adaptation of the type I and type II errors to reflect the potential benefits and risks of the treatment under study. In this paper, we present a novel and tractable Bayesian decision-theoretical framework on multi-period adaptive clinical design that accommodate different forms of experiments (or controls) in each period, and in which the type I and type II errors emerge endogenously as a result of the rewards associated with making the correct decision and penalties from incorrect ones. Our framework builds on the sequential hypothesis testing paradigm in which the clinical trial designer can either choose among different experiments with different information, or to terminate the trial. We show that the log-likelihood ratio (LLR) converges to a diffusion process via a limiting approximation, where the designer runs a series of increasingly less informative experiments whose distributions under the null and the alternative converge in symmetric KL-divergence. The optimal solution to the resulting stochastic control problem is derived analytically. We demonstrate using real-world clinical trial data that compared to the existing non-adaptive policy, our model achieves a 20 - 50\% improvement in overall expected economic benefits, which is a result of higher accuracy in terminal decisions, lower experimental costs and shorter termination time

Keywords: Business.
Advisor: Zenios Stefanos A. 
Wein Lawrence 
Xu Kuang