Applications of Levy Processes to Mathematical Finance

征收过程在数学金融中的应用

• 批准号: 13640131
• 负责人: MIYAHARA Yoshio
• 金额: $ 0.64万
• 依托单位: Nagoya City University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640131/
• 关键词: mathematical finance; option pricing; geometric Levy process; martingale measure; relative entropy; 平滑近似法; Poisson random measure; オプション価格; 確率数値近解析; Markov chain; 確率数値近似法; 時系列データ平滑化; subordination

We have studied the option pricing problems in the incomplete asset market, which is one of the important problems in the field of mathematical finance. Our goal is the construction of the [Geometric Levy process & MEMM] pricing model, in which the geometric Levy processes are adopted as the underlying asset price processes and the MEMM (=minimal entropy martingale measure) is adopted as the martingale measure. And we have investigated the fundamental theories for the construction of this model and the applications of this mode to the option pricing.We first established the existence theorem of MEMM for the geometric Levy processes, and we nest investigated the properties of MEMM and the properties of the [Geometric Levy process & MEMM] pricing model. Especially we have studied the relations between the MEMM and the Esscher martingale measure comparing each other. We investigated the methods for the application of this model, for example the method for the estimation of Levy processes. We also investigated the calibration problems of our model.

本文研究了不完全资产市场中的期权定价问题，这是金融数学领域的重要问题之一。我们的目标是构造[Geometric Levy过程& MEMM]定价模型，其中几何Levy过程被用作基础资产价格过程，MEMM（=最小熵鞅测度）被用作鞅测度。首先建立了几何Levy过程的MEMM存在定理，然后研究了MEMM的性质和[Geometric Levy过程& MEMM]定价模型的性质。特别是通过比较研究了MEMM与Esscher鞅测度之间的关系。我们研究了该模型的应用方法，例如Levy过程的估计方法。我们还研究了我们的模型的校准问题。

项目成果

Y.Miyahara: "A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes"Discussion Papers in Economics, Nagoya City University. No.379. 1-14 (2004)

Y.Miyahara：“A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes”经济学讨论论文，名古屋市立大学。

Tetsuya Misawa: "A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations"SIAM Journal on Scientific Computing. Vol.23. 866-890 (2001)

Tetsuya Misawa：“随机微分方程数值积分的李代数方法”SIAM 科学计算杂志。

A.Shimizu, T.Soshi: "Positively recurrent Markov chains and the stepping stone model as a Fleming-Viot process"Yokohama Mathematical Journal. 49. 89-103 (2001)

A.Shimizu、T.Soshi：“正循环马尔可夫链和作为 Fleming-Viot 过程的垫脚石模型”横滨数学杂志。

T.Mori, T.Misawa: "Analysis of Time Series Data by Smooth Fitting with Meyer Wavelets"Discussion Papers in Economics, Nagoya City University. Vol.344. 1-27 (2003)

T.Mori、T.Misawa：“通过 Meyer 小波平滑拟合分析时间序列数据”经济学讨论论文，名古屋市立大学。

Y.Miyahara: "[Geometric Levy Process & MEMM] Pricing Model and Related Estimation Problems"Asia-Pacific Financial Markets. 8(2001), No.1. 45-60

Y.Miyahara：“[几何征费过程