The Research On Stochastic Numerical Algorithms For Backward Problems

The research of Forward Stochastic Differential Equations(FSDE) began at the end of 1940s,which has direct application background,but also posesses perfect framework of theory.Comparing with FSDE,the research of Backward Stochastic Differential Equations(BSDE) originated in recent 30 years.Although the history is not long,the development is greatly rapid and lots of results are obtained.Now,BSDE has been penetrated in many fields,such as PDE,Financial Mathematics,Stochastic Control, Differential Geometry,Stochastic Game Theory.So that BSDE becomes a powerful maths tool gradually.But until now,most of BSDE' research is qualitative.Altough linear and some special equations' solutions are analytic,the nonlinear BSDE' solutions have no analystic representations.However,a number of scholars have turned to look for the numerical solutions of BSDE,now it becomes such a hot problem that many research groups and individuals begin their study in this fields.This paper just consider a special form of BSDE which is called Forward-Backward Stochastic Differential Equation(FBSDE). Based on the "Four Step Scheme",that is in FBSDE,the backward equation's adapted solution can be determined explicitly by the forward equation's solution,using a certain parabolic PDE system's solution.So we can discretize nonlinear parabolic equation firstly,then the FSDE,last to obtain the numerical solution of the FBSDE.At present, the research results about discretizing FSDE is quatitive,thus,the most important is how to discretize the nonlinear parabolic equation.To some extent,the research of numerical solution about BSDE turns into PDE.The purpose of this paper is to discretize FBSDE,basing on the four step scheme. Therefore,numerical methods for the corresponding parabolic equation should be presented firstly.In traditionally,the numerical methods for parabolic equation are deterministic, but it is difficult to discretize nonlinear equations.And in this paper,a layer method constructed through a stochastic approach is proposed to solve semilinear and quasilinear parabolic equations.And the construction of the layer method mainly lies in two factors,one is the choice of the probabilistic representations of the solution for parabolic equations,the other is the choice of the discrete methods for the corresponding stochastic differentional equations.The layer methods fit well to solve all dimensional parabolic equations,but the traditional deterministic methods do well only when the dimensions of the equations are no more than 3.And the great virtue of the layer methods is that one can just discretize the time variable t,without considering the spatial variable x.Although we also discretize x to execute the algorithm and to reduce the computational volume,the stability of the layer method is intrinsic.In fact,the truncation of spatial variable x does not significantly affect the accuracy of the numerical results obtained,which is to be testified by the construction of the methods and the numerical tests.In the paper,firstly,a layer method is proposed to solve the cauchy problem for a type of qusilinear parabolic equations in the Chapter 2 and 3,where the coherence of the orginal equations and the corresponding differentiated equations is exploited.In Chapter 2,a layer method is constructed by the weak explicitly Euler method.Moreover, in Chapter 3,a type of much more general layer method is constructed.Essentially speaking,Chapter 3 is the extension and improvement of Chapter 2.Then in Chapter 4, we construct the numerical algorithms to solve the FBSDE connected with semilinear and quasilinear parabolic equations respectively,where we exploit a new method to discretize the quasilinear parabolic equations.Further more,we analyze the convergence of the numerical algorithms,and testify the schemes through some examples.The numerical results shows that the accuracy and feasibility of our algorithms.In Chapter 5,we apply the theory of FBSDE and the algorithms of parabolic equations to price Europe Option. Through the numerical example we testify the validities of the theory and the algorithms.