| The research of rational expectation theory began at 1961. John F. Muth firstly proposed the "rational expectation" concept in his article titled’Rational Expectation and the Theory of Price Movements’. In the 60’s, the concept had been applied to the currency market analysis. In the early 70s, Robert Lucas and other economists had made a significant contribution to the rational expectation hypothesis. Lucas criticized the traditional economic policy and put forward the theory of rational expectation which is currently one of most well-known views---"Lucas critique ". In the past ten years, a number of economists also have introduced rational expectation into micro-finance, using mathematical tools to build rational expectation models, and derived economic unity’s expected value of asset prices. This thesis aims to introduce effects of uncertainty into rational expectation models and use forward-backward stochastic differential equations to study a rational expectation model, which is a generalization of the model introduced by Miller and Weller (1995) and Yannacopoulos (2007). Under certain conditions we obtain the expected value of serveral asset prices, the properties of the assets, and the characteristics of the assets.This thesis is divided into the following six chapters.Firstly we present the background of expectation models, as well as the classification. We explain the reason why we should account for effects of uncertainty in economic model and depict the rational expectation model of Miller and Weller (1995), where we present a brief description of their model--a linear saddlepoint system model. One of the great attraction of the model is that the qualitative nature of the assets can be analyzed by a simple geometric model with a phase plane. Yannacopoulos (2007) pointed out that saddle point with relevant content and equilibrium path in geometry will no longer be available for lack of uncertainty. The concept of the phase plane which is a powerful tool in the study of qualitative dynamics in deterministic and non-autonomous dynamical systems is not, at least to the best of our knowledge, readily applicable to the case where uncertainty is introduced to perfect foresight models. So he made use of the forward-backward stochastic differential equations to deal with rational expectation models. We provide examples to illustrate FBSDEs which have been so far successfully applied to a variety of problems of mathematical finance.Chapter 2 firstly gives the rational expectation model of Miller and Weller (1995) and introduces the model of Yannacopoulos (2007) who used the forward-backward stochastic differential equations to study the rational expectations. In this thesis, we apply the Ito integral and the forward-backward stochastic differential equations to study an expection model which is different from that of Yannacopoulos (2007), and prove that the asset must satisfy a second-order nonlinear differential equation. Finally, we solve three specific examples to show that the discount factor together with coefficients of rational expectation model determines the properties of the assets.Chapter 3 proves that the model (3) we mainly discussed is equivalent to the infinite horizon FBSDE (4) under the conditions such as locally square integrable processes xt, yt and with the boundary condition yt bounded a.s.,uriiformly for all t. We make use of martingale, Wiener process and Ito integral to prove the equivalence.In chapter 4, we restrict our study in nodal point solution. Using Ito’s lemma, we shall derive that the solution of typeⅡis the solution of a non-linear elliptic partial differential equations. Two concrete examples are given.In chapter 5, we shall study the type I solution. Using Ito formula to derive that the expectation solution of type I satisfies a second-order partial differential equations, we solve two specific examples.In chapter 6, we summarize the work of this thesis. In order to guarantee the uniqueness of the assets, we perhaps need to impose some strong assumptions on the coefficients of the rational model. Finally, we raise some critical questions and pave the way for our future study. |
