Numerical Methods And Their Error Estimates For Backward Stochastic Differential Equations

Backward Stochastic Differential Equation (BSDE) is a relatively new direction.The linear case first studied by J-M Bismut [9] in 70's can be considered as a generalization of the celebrated Girsanov theorem. The notion of nonlinear BSDEs wasintroduced by Pardoux and Peng [60] in 1990. Independently, Duffie and Epstein [28]introduced stochastic differential utilities in economic models in 1992, as solutions tosome special BSDEs. From then on, various results and applications of BSDEs weredeveloped, including: Reflected BSDEs, Forward and Backward Stochastic Differential Equations, Connections between PDEs and BSDEs, Applications in the StochasticControl, Applications in the Mathematical Finance, Nonlinear Expectations and Nonlinear Martingales, Recursive Utilities and Risk-Sensitive Utilities, Stochastic Differential Geometry, etc.. One can find more information on both theory and applicationin the books of El Karoui-Mazliak [30], Ma-Yong [51], Yong-Zhou [86] and thesurvey paper of El Karoui-Peng-Quenez [33], especially in mathematical finance andstochastic control, for such equations.The existence and uniqueness theorem of BSDE implies that we are able to knowclearly what to do in order to achieve a given goal in the future. But it is still anunresolved issue that how to work out its solution for a specific equation in generalcase. Explicitly solvable BSDEs are rare in practical applications so that computingapproximate solutions of BSDEs becomes highly desired.Compared to the numerical solution of stochastic differential equations, whetherfrom the richness of the results or the degree of difficulty in the algorithm realizationview, BSDE is far behind. There are nothing more than two reasons for this problem: first, there is a fundamental structural difference between the forward stochastic differential equations and backward stochastic differential equations, therefore we can notdirectly apply the existing numerical methods of SDE to solve BSDE; Secondly, fromthe application perspective, the forward stochastic differential equations primarily dealwith problems of how to understand the objective existence of a stochastic process, butbackward stochastic differential equations are mainly concerned with how to make asystem to achieve the desired goals in a random disturbance environment.In the past decades, many scholars have made great efforts in the numerical solution of BSDEs and made a series of results. These numerical methods can be dividedinto two types in accordance with its solving principles: The main focus of the firstkind of algorithms is the numerical solution of a parabolic PDE which is related tothe BSDE. A second type of algorithms works backwards through time and tries totackle the stochastic problem directly. In 2006, Zhao, Chen and Peng [89] proposedtheθ-scheme for numerical solutions of general BSDEs. This method combines thecharacteristics of numerical methods for PDE, explains the high accuracy finite difference method in random environment. In the scheme, deterministic time and spacepartitions are taken to discretize BSDEs, and the Monte Carlo method with some interpolation approximation procedures is used to calculate the conditional mathematicalexpectations, some better results are obtained in the numerical experiments.In this thesis, we mainly study several numerical methods to solve BSDE. Basedon the idea of Zhao, Chen and Peng [89], instead of using the Monte Carlo method,we use the Gauss-Hermite quadrature rule to approximate conditional mathematicalexpectations, we also prove the error estimates for theθscheme. In addition, we presenta new Crank-Nicolson scheme for BSDE and estimate its error. Finally, we propose ahigher-order scheme for BSDE named the Adams scheme, and give its error estimates.In the following, we list the main results of this thesis.Chapter 1: We give a brief introduction to the background and the general ideafor the topics in this thesis. We review the basic concepts of BSDE and the Feynman-Kac formula and summarize some previous works for the numerical solution of BSDE.Chapter 2: We present the error estimates forθscheme. We prove that theθ- scheme is of first-order convergence for generalθ, and in particular, is of second-orderwhenθ=(?). For the parameterθ=1, we obtain that theθ-scheme is of first-order intime for solving the adapted solution (y_t,z_t) of the BSDE (2-1). In the case ofθ= (?),we also derive an error estimate of the scheme for solving z_t of the BSDE (2-1).We refer the following two equations withθ∈[0,1] for solving (?)as theθ-scheme for discretizing the BSDE (2-1),For the error estimates of theθscheme we have the following theorems.Theorem 2.1. Suppose that Assumption 2.1 holds. Let y_t and y~n be solutions of theBSDE (2-1) and theθ-scheme (2-12), respectively. Then for sufficiently small time stepΔt_n, we havewhere C is a constant depending only on T, upper bounds of derivatives ofφand f,and the solution u(t, x) of (2-3).Theorem 2.3. Suppose that Assumption 2.1 holds. Let y~n(n = N,…,0) be thesolution of theθ-scheme (2-12) withθ= (?), and y_t (0≤t≤T) the solution of the BSDE(2-1). Then for sufficiently small time stepΔt_n, we haveTheorem 2.4. Suppose that Assumption 2.1 holds. Let (y~n,z~n) (n = N,…,0) be thesolution of theθ-scheme (2-12) and (2-13) withθ=(?), and (y_t,z_t)(0≤t≤T) the exact solution of the BSDE (2-1). Then for sufficiently small time stepΔt_n, we haveA fully discreteθ-scheme is defined as follows: Given the random variable y_i~N,i∈Z, find an approximate solution (?) satisfyingAbout the error of the fully discrete scheme we haveTheorem 2.7. Let (y_t, z_t) be the solution of the BSDE (2-1) and (y_i~n, z_i~n) the solutionof the fully discrete scheme (2-70) and (2-71) with the linear polynomial interpolationused to calculate (?). Then under Assumption 2.1, it holdsfor the fully discreteθ-scheme with generalθ∈[0,1]. In particular, we have thatfor the fully discreteθ-scheme withθ=(?), andfor the fully discreteθ-scheme withθ=1.Chapter 3: We propose a new Crank-Nicolson scheme for the general multidimension BSDEs and give its error estimates. We prove that the scheme is of second- order convergence and provide some numerical tests both for theθscheme in Chapter2 and the Crank-Nicolson scheme in this chapter.We call the following two equations in the component form for solving (y~n, z~n)(n =N, N - 1,…, 0) the Crank-Nicolson scheme for the BSDE (3-1),for n = N,N-1,…,0 with the terminal conditions (?) and (?). Here (?), and (?) is the j-thcolumn of the matrix (?).We rewrite them in the following matrix formFor the error estimates of Crank-Nicolson scheme we have the following theo-rems.Theorem 3.1. Suppose that Assumption 2.1 holds. Then for sufficiently small timestepΔt_n, we have the error estimatewhere y_t and y~n are the solutions of the BSDE (3-1) and the Crank-Nicolson scheme(3-18) and (3-19), respectively, C is a constant which depends only on T, upper boundsof derivatives of the functionsφ, f and the solution u(t, x) of the problem (3-4).Theorem 3.2. Let z_t and z~n be the solutions of (3-15) and (3-21), respectively. Suppose that Assumption 2.1 holds, then for sufficiently small time stepΔt_n we haveThe fully time-space Crank-Nicolson discrete scheme is: Find (?), such that (y_i~n, z_i~n) satisfiesChapter 4: We propose the Adams scheme for general BSDE, present the error estimates for BSDE when the generator f is independent of z and prove the highaccurate convergence of the Adams scheme.We call the following two equations the Adams scheme for discretizing BSDE(4-1),for n = N - m,…,0 with the terminal condition y~N given by the terminal condition ofthe BSDE (4-1).The fully discrete Adams scheme is defined as follows: Given the random variabley_i~N, first to solve (y_i~n, z_i~n), i∈Z, n = N - 1,…,N - m + 1 by the Runge-Kutta method,and then find an approximate solution (y_i~n, z_i~n)(n = N - m,…,0,i∈Z) satisfying The error estimates of the Adams scheme for BSDEs with the generator not depending on z are given in the following theorems.Theorem 4.1. Suppose Assumption 2.1 hold and (?),thenfor sufficiently small time step, we have the error estimatewhere y_t and y~n are the solutions of the BSDE (4-1) and the Adams scheme (4-25) and(4-26), respectively, C is a constant, which depends only on T, the upper bounds of thederivatives of the functionsφ, f and the solution u(t, x) of the problem (4-3).Theorem 4.2. Let z_t and z~n be the solutions of (4-23) and (4-25), respectively. SupposeAssumption 2.1 hold and (?),then for reasonable small timestep we have...