The Closed Forms Of The Future Asset In A Rational Expectation Model

Provided that the economic situation possesses certain conditions, economic models of rational expectation exist some equilibriums with saddlepoint structure. Taking account of uncertainty, we know that phase plane is not applicable to analyze these situations. However, the forward-backward stochastic differential equations can be effectively employed to solve these problems. The forward-backward stochastic differential equations have been used in mathematical finance for many years. The strong link with control theory and possibility of alternative formulation makes forward-backward stochastic differential equations more useful in finance.In this paper, we get conclusions that partial differential equations which equal to forward-backward stochastic differential equations have solutions under some special conditions. The exact solutions for the partial differential equations are obtained by using the methods which are widely used in mathematical physics. The corresponding economic meanings of the solutions are explained. In the first chapter, the background and analyzing tools are presented. Chapter 2 introduces linear differential saddlepoint system and non-linear saddlepoint system. It is shown that the forward-backward stochastic differential equations are equivalent to a second-order partial differential equation. The closed forms of the partial differential equation are acquired in chapter 3 by using mathematical methods. The stability of rational assets is investigated in chapter 4 under certain circumstances.