北京师范大学硕士生开题报告样本07年5月-北京师范大学数学科学 - 图文

三、论文大纲 本学术论文预计最终将包含以下内容: 首先是全文的综述, 我们将对论文的背景和动机做简要的说明并概述文章的主要结果. 第一章共分三节: 第一节我们利用一类Poisson随机测度驱动的随机方程构造离散状态的催化分枝模型,具体包括催化DB过程与催化DBI-过程. 第二节我们利用一类白噪声和Poisson随机测度驱动的随机方程构造了一类新的催化CBI-过程. 第三节我们首先证明若干重要的命题, 主要涉及‘重整化’概率母函数的极限函数的表示结果; 在此基础上证明上述催化CBI-过程可由一列催化DBI-过程经重整化依轨道弱收敛得到. 第二章共分三节: 第一节对仿射过程作了一个简要的陈述, 并列出了相关的生成元的刻画. 第二节研究了催化DBI-过程的高密度波动极限定理及其逆定理. 第三节我们讨论了催化CBI-过程的小分枝低密度波动极限定理. 第三章我们证明了二维情形的CBI-过程可由两物种GWI-过程经自然重整化后依轨道弱收敛得到, 这扩展了Li (2005).的部分结果, 并进一步放宽了定理的条件. 四、主要参考文献 1. Aldous, D.: Stopping times and tightness. Ann. Probab. 6 (1978), 335-340. 2. Athreya, K.B. and Ney, P.E.: Branching processes. Spring-Verlag, New York (1972). 3. Asmussen, S. and Hering,H.: Branching Processes. Birkhauser, Basel (1983). 4. ?inlar, E. and Jacod, J.: Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. Seminar on Stochastic Processes. (1981). 5. Cox, J., Ingersoll, J. and Ross, S.: A theory of the term structure of interest rates. Econometrica 53 (1985), 385-408. 6. Dawson, D.A. and Fleischmann, K.: A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 (1997), 213-276. 7. Dawson, D.A. and Fleischmann, K.: Catalytic and mutually catalytic branching. Infinite dimensional stochastic analysis. (2000), 145--170. Royal Netherlands Academy of Arts and sciences, Amsterdam. 8. Dawson, D.A.; Li, Z.H.; Wang, H.: Superprocesses with dependent spatial motion and general branching densities. Electronic Journal of Probability. 6 (2001), 1-33. 9. Dawson, D.A. and Fleischmann, K.: Catalytic and mutually catalytic super-Brownian motions. Progr. Probab. 52 (2002), 89--110. Birkh?user, Basel. 10. Dawson, D.A.; Li, Z.H.: Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions. Probab. Theory Related Fields. 127 (2003), 37-61. 11. Dawson, D.A.; Li, Z.H.; Wang, H.: A degenerate stochastic partial differential equation for the purely atomic superprocess with dependent spatial motion. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 6 (2003), 597-607. 12. Dawson, D.A. and Li, Z.H.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34 (2004), 1103-1142. < 第 5 页 共 7 页 > 研究生院培养处制表

13. Duffie, D.; Filipovi?, D. and Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13 (2003), 984-1053. 14. Ethier, S.N. and Kurtz, T.G.: Markov processes: Characterization and Convergence. John Wiley and Sons Inc., New York (1986). 15. Fang, S. and Zhang, T.: A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields. 132 (2005), 356-390. 16. Feller, W.: Diffusion processes in genetics. University of California Press, Berkeley and Los Angeles, (1951). 17. Grey, D.R.: Asymptotic behavior of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 (1974), 669-677. 18. Gorostiza, L.G.; Li, Z.H.: High density fluctuations of immigration branching particle systems. Stochastic Models CMS Conference Proceedings 26 (2000) 159-171. 19. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (1993), 327-343. 20. Ikeda, N. and Watanabe, S.: Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland/Kodansha, Amsterdam/Tokyo (1989). 21. Ji?ina, M.: Stochastic branching processes with continuous state space. Gzechoslovak. Math. J. 8 (1958), 292-313. 22. Karoui, N.and Méléard, S.: Martingale measures and stochastic calculus. Probab. Theory Related Fields. 84 (1990), 83--101. 23. Klenke, A.: A review on spatial catalytic branching. In: CMS Conference Proceedings. Series 26 (2000), 245--263. Amer. Math. Soc., Providence, RI. 24. Kawazu, K. and Watanabe, S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 (1971), 36-54. 25. Jocod, J. and Schiryaev, A.N.: Limit theorems for stochastic processes. Grundlehren der mathematischen Wissenschaften, vol. 288. Springer-Verlag, Berlin-Heidelberg-New York (1987). 26. Lambert A.: The branching process conditioned to be never extinct (unpublished). [Preprint from: biologie.ens.fr/ecologie/ecoevolution/lambert] 27. Le Gall, J.F. and Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998), 213-252. 28. Le Gall, J.F. and Le Jan, Y.: Branching processes in Lévy processes: Laplace funtionals of snake and superprocess. Ann. Probab. 26 (1998), 1407-1432. 29. Li, Z.H.: Integral representations of continuous functions. Chinese Sci. Bull.} (Chinese Ed.) 36 (1991), 81-84 / (English Ed.) 36 (1991), 976-983. [Preprint from: math.bnu.edu.cn/\\~lizh] 30. Li, Z.H.: Asymptotic behavior of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 (2000), 68-84. 31. Li, Z.H.: Ornstein-Uhlenbeck type processes and branching processes with immigration. J. Appl. Probab. 37 (2000), 627-634. 32. Li, Z.H.; Wang, H.; Xiong, J.: A degenerate stochastic partial differential equation for superprocesses with singular interaction. Probability Theory and Related Fields. 130 (2004), 1-17. 33. Li, Z.H.; Zhang, M.: Fluctuation limit theorems of immigration superprocesses with small branching. Statistics and Probability Letters 76 (2006), 401-411. 34. Li, Z.H.: A limit theorem of discrete Galton-Watson branching processes with immigration. J. Appl. Probab. 43 (2005), 289-295. 35. Li, Z.H.: Branching processes with immigration and related topics. Frontiers of Mathematics in China, Selected Publications from Chinese Universities. 1 (2006)1: 73-97. < 第 6 页 共 7 页 > 研究生院培养处制表

36. Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Related Fields. 123 (2002), 157--201. 37. Pagliaro, L. and Taylor, D. L.: Aldolase exists in both the fluid and solid phase of cytoplasm. J. Cell Biology. 107 (1988), 981-999. 38. Pakes, A.G.: Some limit theorems for continuous-state branching processes. J. Austral. Math. Soc. Ser. A 44 (1998), 71-87. 39. Pakes, A.G.: Revising conditional limit theorems for mortal simple branching processes. Bernoulli 5 (1999), 969-998. 40. Sato, K.: Lévy processes and Infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics. 68. Cambridge University Press, Cambridge (1999). 41. Shiga, T. and Watanabe, S.: Bessel diffusion as a one-parameter family of diffusion processes. Z. Wahrsch. verw. Geb. 27 (1973), 37-46. 42. Taira, K.: Diffusion processes and Partial differential equations. Academic Press. London (1988). 43. Venttsel', A. D.: On boundary conditions for multi-dimensional diffusion processes. Theor. Probability Appl. 4 (1959), 164—177. 44. Wang, H.: A class of measure-valued branching diffusions in a random medium. Stochastic Anal. Appl. 16 (1998), 753—786.

五、导师意见 催化分枝过程和仿射过程是两类新的随机过程模型, 前者起源于生物学和化学模型的研究, 后者主要来源于数理金融. XXX的论文选题在于考虑催化分枝过程在不同重整化之下的轨道弱收敛问题, 并建立这种分枝过程与仿射过程之间的联系. 该选题有很强的前沿性, 预期的结果有重要的理论意义和应用价值. 所以同意该选题. 导师签名： (李增沪) 2005年12月20日

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