Tauberian and Mercerian theorems and analysis of stochastic financial processes

陶伯里安和默塞里安定理以及随机金融过程分析

• 批准号: 14540147
• 负责人: INOUE Akihiko
• 金额: $ 2.24万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540147/
• 关键词: Tauberian theorem; fractional Brownian motion; partial autocorrelation function; predictor coefficient; expected utility maximization; innovation process; financial market model; タウバー型過程; フラクショナル・ブラウン運動; フラクショナル・ブウウン運動; 予測理論; 長時間記憶; 記憶を持つ資産過程

We introduced a generalized fractional Brownian motion and proved a finite-past prediction formula for it using Tauberian theorems and a new prediction-theoretic approach.By applying a new prediction-theoretic approach to stationary time series, we proved a surprisingly simple representation theorem for the partial autocorrelation function Using the result, we derived a precise asymptotic behavior with remainder for it.We introduced a class of stationary increments processes which are described by continuous-time AR-equations of infinite order. We also considered SDE with the stationary increments process as driving force. In this way, we introduced stock price processes with long or short memory. We discussed about completeness and behavior of volatility of the financial market with this stock price process. We obtained an explicit representation of the innovation process associated with the drinving force, using a new prediction-theoretic approach. In this way, we solved the problem of expected utility maximization problem in the financial market. In the simplest case, this is a paremetric model that has only two additional parameters compared with the Black-Scholes model. We observed that this model well describes the real market data such as S & P500. This shows the usefulness of the model.

我们引入了广义分数布朗运动，并使用陶伯定理和新的预测理论方法证明了它的有限过去预测公式。通过将新的预测理论方法应用于平稳时间序列，我们证明了一个令人惊讶的简单表示定理偏自相关函数 利用结果，我们推导了带有余数的精确渐近行为。我们引入了一类由无限阶连续时间 AR 方程描述的平稳增量过程。我们还考虑以平稳增量过程作为驱动力的 SDE。这样，我们就引入了具有长记忆或短记忆的股票价格过程。我们通过这个股票价格过程讨论了金融市场波动的完整性和行为。我们使用新的预测理论方法获得了与驱动力相关的创新过程的明确表示。这样，我们就解决了金融市场中的期望效用最大化问题。在最简单的情况下，这是一个参数模型，与 Black-Scholes 模型相比，仅具有两个附加参数。我们观察到该模型很好地描述了 S & P500 等真实市场数据。这显示了该模型的实用性。

项目成果

T.Mikami: "Monge's problem with a quadratic cost by the zero-noise limit of h-pathprocesses"Probab.Theory Related Fields. (発表予定).

T.Mikami：“蒙日的 h 路径过程的零噪声极限的二次成本问题”Probab.Theory 相关领域（待提交）。

A.Arai: "Non-relativistic limit of a Dirac-Maxwell operator in relativistic quantum electrodynamics"Rev.Math.Phys.. 15. 245-270 (2003)

A.Arai：“相对论量子电动力学中狄拉克-麦克斯韦算子的非相对论极限”Rev.Math.Phys.. 15. 245-270 (2003)

A.Arai: "Non-relativistic limit of a Dirac-Maxwelll operator in relativistie quantum electrodynamics"Rev.Math.Phys.. 15. 245-270 (2003)

A.Arai：“相对论量子电动力学中狄拉克-麦克斯韦算子的非相对论极限”Rev.Math.Phys.. 15. 245-270 (2003)

A.Arai: "Enhanced binding in a general class of quantum field models"Rev.Math.Phys.. 15. 387-423 (2003)

A.Arai：“增强了一般类量子场模型中的结合”Rev.Math.Phys.. 15. 387-423 (2003)

T.Mikami: "Monge's problem with a quadratic cost by the zero-noise limit of h-path processes"Probab.Theory Related Fields. (発表予定).

T.Mikami：“蒙日的 h 路径过程的零噪声极限的二次成本问题”Probab.Theory 相关领域（待提交）。