An Asymptotic Expansion Approach to Numerical Problems in Finance

1.给出了马尔可夫和非马尔科夫环境下未定权益估值的渐近展开方法的数学有效性。(《关于渐近展开方法在未定权益分析中的有效性》，《应用概率年鉴》(2003)。)2.当状态变量服从一般扩散过程时，我们给出了动态投资问题中最优投资组合的数值计算公式。此外，我们还给出了基于渐近展开方法的解析近似公式，并通过数值算例说明了其有效性。(《最优投资问题的渐近展开方案》，即将在随机过程的统计推断中发表。)3.我们证明了当标的价格服从一般扩散过程时，美式期权的价值被分解为相应的欧式价值和早期的行权溢价。然后，我们应用渐近展开方法发展了一种半解析计算格式。(《美国期权的渐近展开方法》，《货币与经济研究》(2003)。)4.我们提出了一种新的计算格式，利用渐近方法来实现金融数值分析中蒙特卡罗模拟的方差化。给出了该方法的一般方案和数学有效性。应用实例包括跳跃扩散过程下的期权定价、HJM框架下的利率衍生品定价以及动态投资问题中的最优投资组合计算。(《基于Malliavin-Watanabe演算的渐近展开方法在金融问题中的应用》，即将出版的《随机过程和数学金融的应用》。)5.我们出版了一本关于金融的渐近展开方法的书(《数学金融的基础-Malliavin演算和渐近展开方法的应用--》，(2003)。)

1.We developed mathematical validity of the asymptotic expansion approach to valuation of contingent claims in Markovian and non-Markovian setting. ("On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis," Annals of Applied Probability(2003).)2.We derived numerically tractable formulas of optimal portfolio in the dynamic investment problems when state variables follow general diffusion processes. Moreover, we provided analytic approximation formulas based on the asymptotic expansion approach and showed their effectiveness through numerical examples. ("An Asymptotic Expansion Scheme for the Optimal Investment Problems," forthcoming in Statistical Inference for Stochastic Processes.)3.We showed a decomposition of the value of an American option into the corresponding European value and the early exercise premium when the underlying price follow a general diffusion process. Then, we applied the asymptotic expansion approach to develop a semi-analytic computational scheme. ("An Asymptotic Expansion Approach to American Options," Monetary and Economic Studies(2003).)4.We proposed a new computational scheme with the asymptotic method to achieve variance reduction of Monte Carlo simulation for numerical analysis in finance. We provided general scheme and mathematical validity of our method. The examples of the application include pricing options under jump-diffusion processes, pricing interest rate derivatives in HJM framework and computing optimal portfolios in the dynamic investment problems. ("Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems," forthcoming in Stochastic Processes and Applications to Mathematical Finance.)5.We published a book on the asymptotic expansion approach to finance.("Foundation of Mathematical Finance-Application of Malliavin Calculus and Asymptotic Expansion Method-," (2003).)