The stochastic incentive effect of venture capital in partnership systems with the asymmetric bistable Cobb-Douglas utility

Partnerships, between multiple sides that share cooperative goals, strive for mutual benefit, and acknowledge a high level of mutual interdependence, are ubiquitous both between and within the enterprises, and the internal or external stochastic factors driving competition and cooperation are the fundamental characteristics of partnership systems. Thus motivated, we establish an over-damped Langevin equation to describe the stochastic dynamical behaviors of the enterprise subject to asymmetric bistable Cobb-Douglas utility (CDU) potential. Due to the contemporaneous presence of periodic capital-product switches and stochastic fluctuations of internal and external capital environment, the stationary response of partnership systems is driven by the combination of the two driving effects cooperatively cause the enterprise to switch between the two utility equilibriums, and produce the maximum of stochastic incentive effect in the statistical sense. Based on the two-state theory, we derive the analytical results of performance measurement, including output signal-to-noise ratio (SNR), stationary unit risk-return (URR) and the incentive risk, which are divided into two categories: systematic risk and bilateral risk. Finally, one true example are introduced, and our proposed model is used to fitly explain the 'U'-shape phenomenon observed from small and medium-sized enterprise (SME) samples. The purpose in this paper is to develop a quantitative method and the associated prototype system try to answer the questions of how the venture capital incents the partners especially associated with partnership success, what roles the internal and external risks play respectively, and how to avoid risk resonance and create portfolio strategies of introducing venture capital and optimizing the portfolio risk.