Estimation Of Parameters In Forward-backward Stochastic Differential Equations

Recent years have seen an increasing number of research about forward-backward stochastic differential equations(FBSDEs),meanwhile,only several of reference documents are found about the work of estimating parameters in FBSDEs.There are only references[5,7,8,9,10]aimed to estimate parameters in FBSDEs,and references[4,11,12,13,14,15]are close to this aim.After reading all the references,we find 4 problems as follow worth our further study.(1)Nonparametric estimation g of FBSDEs in general condition given by existing literature has only two types,ght,(s)and ght,hx(s,x).This bases on the relationship between the solution of FBSDE(Y_t,Z_t)and viscosity solution of a specific partial differential equation(u,v),Y_t = u(t,Xt),Z_t =v(t,Xt),that is to say,(Y_t,Z_t)also depends on(t,Xt).So when we only need to evaluate g(s,x,y,z)and the actual formulas of g(s,x,y,z)isn't necessary,g(s,x,y,x),z(s,x))depends on(s,x)only.But in this paper,we think the aim of estimating the generator g(s,x,y,z)is to calculate g(s0,x0,y0,z0)when arbitrary(s0,x0,y0,z0)was assigned.In order to solve this problem,we try to give a specific expression of g(s,x,y,z)in the main body of this paper.(2)There is only one specialized form of g is covered in existing litera-ture,that is linear form.In consideration of the significance in practical appli-cation,we investigate the estimation methods and numerical simulation in the situation of g=u|Z_t| especially.And we give the expression of the bias and the process of proof referring to the methods in existing literatures.(3)In the section of numerical simulation in existing literatures,the selec-tion of nonparametric regression methods and the selection of bandwidth are less noticed,so is the stability of estimators.We will investigate them further in the section of numerical simulation of this paper.(4)In existing literatures,litter effort has been put.into applying practical data,so this paper tries to apply FBSDE models into empirical analysis of the 50 ETF options in Shanghai Stock Exchange.We will inspect the application value of the estimation methods for parameters in FBSDEs in solving the problem of option pricing.We will solve the 4 problems above in the body of this paper,and the arrangement of chapters and sections is as follow.In Chapter 1 we simply introduce preliminary knowledge used in this pa-per,including the introduction of object of our study(FBSDE models)and the introduction of the methods used(nonparametric regression method).In §1.1 we introduce the correlated theory of FBSDE models,including the definition of FBSDEs,the examples of FBSDEs,the types of FBSDEs,the development of F-BSDEs and the differences between FBSDE,BSDE,SDE and ODE.And in §1.2 we organize theory knowledge of nonparametric regression,briefly describing the common methods of nonparametric density estimation and nonparametric regression estimation.In §2.1 we explain the purpose of estimating parameters in FBSDEs and the research value.In §2.2 the literature review about estimation for parameters in FBSDEs is displayed to explain the current situation of study.In §2.3 we point out our innovation in theory and in practice,and we list our supplements.In.Chapter 3,we introduce the estimation of parameters in FBSDE of general form.From §3.1 to §3.4,in order to illustrate the key points of estima-tion methods,we successively introduce the derivation of expressions for es-timators,asymptotic property,terminal condition embed and so on.In §3.5 we summarize the methods and generalize a procedure for estimating process.In§3.6,we give a further description of the expressions for estimators,and this is a solution to problem 1.In Chapter 4,we investigate the estimation methods in the situation of g = u|Z_t|,and give a solution to problem 2 by referring to the methods in existing literatures.In Chapter 5,we conduct numerical simulation.And this chapter covers the solution to problem 3.In Chapter 6,we apply FBSDE models to the price sequence of the 50 ETF options in Shanghai Stock Exchange to solve problem 4.In Chapter 7,we summarize our research results briefly and make an out-look for the direction of future research.