| Abstract | 第1-10页 |
| 摘要 | 第10-14页 |
| 1 Background | 第14-17页 |
| ·Backward stochastic differential equations | 第14-15页 |
| ·Nonlinear Feynman-Kac formula | 第15-17页 |
| 2 A Simple Review of Numerical methods for solving BSDEs | 第17-32页 |
| ·Four step scheme | 第18-20页 |
| ·Delarue and Menozzi's method | 第20-22页 |
| ·Binomial random walk method | 第22-25页 |
| ·Scheme 1 | 第23-24页 |
| ·Scheme 2 | 第24-25页 |
| ·Quantization method | 第25-27页 |
| ·Least-squares regression method | 第27-29页 |
| ·θ-Scheme | 第29-32页 |
| 3 Multi-Step Scheme for solving BSDEs | 第32-49页 |
| ·Preliminaries | 第32-33页 |
| ·Reference Equations for the Multi-step Scheme | 第33-38页 |
| ·The case of m = d = 1 | 第34-36页 |
| ·The general case | 第36-38页 |
| ·A Stable Multi-step Discretization Scheme | 第38-42页 |
| ·The semi-discrete scheme | 第38-40页 |
| ·The fully discrete scheme | 第40-42页 |
| ·Error Estimates of the Multi-step Semi-discrete Scheme | 第42-49页 |
| 4 An efficient Scheme by Using Gauss-Hermite Process | 第49-64页 |
| ·Some properties of the time-space domain | 第49-51页 |
| ·Gauss-Hermite process | 第51-54页 |
| ·An Efficient scheme for solving BSDEs | 第54-56页 |
| ·Error Estimates of the Efficient Scheme | 第56-62页 |
| ·Parallelization | 第62-64页 |
| 5 Numerical Experiments | 第64-83页 |
| ·The effectiveness of the multi-step scheme | 第65-74页 |
| ·Example 1: a BSDE with f(t,y_t) being independent of z_t | 第65-67页 |
| ·Example 2: a BSDE with nonlinear f(t,y_t, z_t) | 第67-68页 |
| ·Example 3: a BSDE with two independent Brownian motions | 第68-69页 |
| ·Example 4: a BSDE with two equations | 第69-71页 |
| ·Example 5: the BSDE derived from Black-Scholes Equation | 第71-74页 |
| ·The effectiveness and efficiency of the scheme by using Gauss-Hermite process | 第74-81页 |
| ·Gauss-Hermite process v.s. Gauss-Hermite quadrature rule | 第74-78页 |
| ·Gauss-Hermite process v.s. multinomial trees | 第78-81页 |
| ·Parallelization Performance | 第81-83页 |
| Bibliography | 第83-88页 |
| CURRICULUM VITAE | 第88-89页 |
| 致谢 | 第89-90页 |
| 学位论文评阅及答辩情况表 | 第90页 |
