A Study Of Continuous-time Contract Under Uncertainty

The study of contracting is one of the most basic and extensive issues in modern economics.The uncertainty of the model,the uncertainty of market and risks caused by incomplete information of the principal and the agent will affeet the design and implementation of contracts.In order to solve the default risk caused by the information asymmetry between the principal and the agent,this paper assumes that the agent does not have access to financial markets,while the principal has access to financial markets,and sets the participation constraint to prevent the default risk caused by the uncertainty of the agent.Earlier studies on contract theory did not take the uncertainty of the model,and this paper takes the uncertainty of Knight into account in the contract model.Firstly,in the case of Knight uncertainty,the agent gives the income to the principal,and the principal returns the consumption to the agent and obtains the utility.Besides,it obtains the G-HJB equation which is satisfied with the value function of the maximum principal utility.Secondly,by using a duality method,Lagrangian function and the duality theorem of weak and strong,the explicit solution of the original value function is obtained by solving the dual value function.Finally,presenting an example of agent income following G-Brownian motion stochastic differential equation,the results of the example are simulated numerically,combining with the uncertainty of Knight,the economic analysis is carried out.The conclusion shows that while the initial promised value to the agent is small while the agent endowment is constant.However,the principal will make a loss when the principal's initial commitment to the agent is large enough.What is more,the Knight uncertainty will not affect the overall change trend of the agent's utility.On the other hand,in real life,when the client and agent sign the contract,they also have to face the uncertainty brought by the inflation of the market.'It can be found from many studies that the inflation uncertainty has a significant impact on the consumption and income of agents.So this paper takes the inflation uncertainty into the continuous time contract problem.First of all,the agent gives the income to the principal,and the principal returns the consumption to the agent and derives utility,after discounting the agent's income and the agent's consumption by inflation,the contract is designed with the goal of maximizing the principal's utility,and the HJB equation is satisfied with the value function of the principal's maximum utility.Secondly,adopting Lagrange function and the duality theorem of weak and strong,the explicit solution of the orig:inal value function is obtained by solving the dual value function.Finally,an example of a utility function of u(c)=c?/P is proposed under inflation,the results of the example are numerically simulated by MATLAB,and the economic analysis is carried out in combination with inflation.The conclusion shows that the utility of the principal while the initial promised value of the principal to the agent is increased.The principal will get profits when the principal's initial promised value to the agent is small.When the principal's initial promised value to the agent is large enough,the principal will get loses.Moreover,neither the in:flation factor ?L nor ?L will affect the overall trend of change in the principal's utility.This paper expounds the background and significance of the research,combing the current research status of contract theory,Knight uncertainty and inflation at home and abroad,summarizing the research results of this paper,and giving the deficiencies and aspects that can be improved in this paper.