Numerical Solutions For FBSDEs And Their Applications In Finance

As the modern society developing,the financial products become more and more important in people's lives,and the mathematical models to analyze the financial market also emerge in an endless stream.Forward-Backward Stochastic Differential Equations are kind of very important stochastic differential equations which developed in 1990s.As the Backward Sto-chastic Differential Equations play more and more important role in financial mathematics, the Forward-Backward Stochastic Differential Equations also show their importance in dealing financial problems.In this thesis,we will discuss the numerical theories for the Forward-Backward Stochastic Differential Equations and their applications in financial problems.In the introduction of this thesis,we introduce the history of financial market and derived financial products,the development of financial mathematic theories.And we also tell the role which Backward Stochastic Differential Equations plays in the development of financial mathematics.In the second chapter of this thesis,we introduce some basic theories of Backward Stochastic Differential Equations and Forward-Backward Stochastic Differential Equations.We also show their relationship to Partial Differential Equations.The third chapter tells a series of financial mathematic models.And in this chapter,we will show the great role which the Backward Stochastic Differential Equations play in the calculation and analysis of financial trading strategy.In the aspect of the relationship between Forward-Backward Stochastic Differential Equations and finance,we will introduce a financial problem which is called the "large investor's trading strategy" and the model which could solve this problem.Chapter 4 discuss the numerical methods for solving Forward-Backward Stochastic Differential Equations.In this thesis,we use the similar method as the 0 scheme,discretize the equations on time-space discrete grids and try to solve the equations in the backward direction. While we dealing with Forward-Backward Stochastic Differential Equations,we need to solve nonlinear equations.In order to simplify the solving process and reduce the time of calcultion, we use n-ary tree to simulate brownian motions and then get estimations of conditional mathematic expectations while we solving the equations.The last chapter of this thesis provides some calculating examples in which we solve the Forward-Backward Stochastic Differential Equations with the method mentioned in the thesis. We first letθbe 0.5 in theθscheme,and use binary tree,ternary tree and quadtree to calculate the examples.For further comparison,we letθ= 1.0 again and process another calculation. At last,we will discuss the error and compare the convergence rates.