An Asymptotic Expansion Approach to Numerical Problems in Finance

1.在马尔可夫和非马尔可夫背景下，我们证明了渐近展开法在未定权益定价中的数学有效性。（“论渐近展开法在或有索赔分析中的有效性”，《应用概率年鉴》（2003年）。）2.在状态变量服从一般扩散过程的动态投资问题中，给出了最优投资组合的数值计算公式。此外，我们提供了基于渐近展开方法的解析近似公式，并通过数值例子表明其有效性。（“An Asymptotic Expansion Scheme for the Optimal Investment Problems”，即将发表于Statistical Inference for Stochastic Processes。3.当标的资产价格服从一般扩散过程时，我们将美式期权的价值分解为相应的欧式期权价值和提前执行溢价。然后，我们应用渐近展开方法来开发一个半解析计算方案。（“An Asymptotic Expansion Approach to American Options，”Monetary and Economic Studies（2003）.）4.提出了一种新的计算方案，利用渐近方法实现金融数值分析中Monte Carlo模拟的方差缩减。给出了该方法的一般方案和数学上的有效性。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微积分的渐进展开方法在金融问题中的应用”，即将发表在《随机过程与数学金融应用》中。）5.我们出版了一本关于金融的渐近展开方法的书。（“金融数学基础--Malliavin微积分与渐近展开法的应用--”，（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).)