The Study Of Several Problems For Rational Expectation Models

Economic crisis of 1970s in western countries has become the crisis of Keynes’s economic theories. They can not explain the phenomenon that economic depression, unemployment and inflation coexisted. It is also an opportunity for Rational Expectation School to develop. Some economists believe that rational expectation theory is one of the important theories of macroeconomics. The theory has certain influence to the method of macroeconomic analysis and theory structure.The so-called expectation is that economic activities in the future are forecasted or estimated by economic trends and economic variables in order to guide people’s decisions and behaviors. The concept of rational expectation first appeared in John·Muth’s paper, "Rational Expectations and the Theory of Price Movement". He thinks that people’s expectation tends to the theoretical prediction results under the same information context. The study of the expectation will help us to judge the main behavior patterns of economic activities.The structure of saddle point equilibrium exists in some of the models that do not introduce uncertainty or the expected model that the current economic literature refers to. But a realistic economic model must necessarily account for the effects of uncertainty. We use forward-backward stochastic differential equations (FBSDEs) in this paper. It is a system of stochastic differential equations, part of which is treated forward in time and part of which is treated backward in time. It has been so far applied to a variety of problems of mathematical finance. The basic problem is that of contingent claim pricing, where a derivative product whose price depends on an underlying asset is to be valued.In this paper a generalized continuous time rational expectation model is re-formulated as forward-backward stochastic differential equations (FBSDEs). Under certain assumptions, the properties of solutions for the model are established. We give several specific examples to illustrate that the discount factor of rational expectations model together with the coefficient of the model equation determines the characteristics of the assets. Finally, we study the compatibility of assets in the rational expectation model.Chapter one describes the development of rational expectation theory, the tools and research methods which are used by other researchers. We state the application of forward-backward stochastic differential equations in economy, finance and a number of related research areas. Chapter two focuses on showing the stochastic saddle-point system, including the linear stochastic saddle-point structure and nonlinear stochastic saddle-point structure. A generalized continuous time rational expectation model is reformulated as forward-backward stochastic differential equations (FBSDEs). In Chapter three, applying the Ito’s formulae and the FBSDEs, we show that the assets in the model must satisfy a nonlinear partial differential equation under certain assumptions. Concrete examples are provided to illustrate the changes of the assets in Chapter four. In Chapter five, we study the compatibility of expected assets.