正倒向随机微分方程与两类衍生模型的统计推断及金融中的应用

项目名称： 正倒向随机微分方程与两类衍生模型的统计推断及金融中的应用

项目编号： No.11501314

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 张齐

作者单位： 青岛大学

项目金额： 18万元

中文摘要： 随着现代社会金融研究领域的日益深入，各种金融产品的数理模型和分析工具层出不穷，如Merton模型等，上世纪末频发的金融危机也凸现这类研究的巨大意义。倒向随机微分方程与正倒向随机微分方程正是在此大环境下被发掘出旺盛的生命力，这类模型已被广泛应用于数理金融与生物动力系统等领域，它依赖于终端条件的本质特征恰好适应某些金融市场或生态环境的运行态势，然而终端相依模型的统计推断工作仍处于起步阶段。本项目构建正倒向随机微分方程模型的贝叶斯推断，包括单一风险和多风险投资场合下模型的贝叶斯估计及其理论性质，模拟实证及方法对比分析；利用贝叶斯变量筛选和测量误差模型似然变量选择提取最优投资组合；另外依据市场需求建立两类正倒向衍生模型，首先是混合期权定价模型，基于混合专家分级法进行推断，验证其良好的样本拟合表现，分析混合权重描述的候选模型偏好程度；其次建立带约束的时变模型，提出拟工具变量回归，实现模型可识别纠偏。

中文关键词： 正倒向随机微分方程；贝叶斯推断；变量选择；混合专家分级法；工具变量

英文摘要： With the development of modern society and the deepening of financial field, financial products have become the indispensable part of people's life, and various mathematical models and analysis tools for dealing with the financial market transactions, which including investment portfolio analysis, asset pricing and financial risk measure, also emerge in an endless stream. Since the nineteen nineties, the worldwide financial crises happened frequently have highlighted the significance of such research to prevent these disasters. Backward stochastic differential equation has been well developed in financial mathematics, and Forward-backward stochastic differential equations is also playing its increasingly important role. Backward Stochastic Differential Equation (BSDE) has been well studied and widely applied in mathematical finance. The main difference from the original stochastic differential equation is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inferences for such a model has hardly been explored in the existing literature. The subject will propose the terminal-dependent bayesian statistical inferences of Forward-Backward Stochastic Differential Equation (FBSDE), including both the single and the multiple risk investment cases, and their properties are discussed by theory and simulation study simultaneously; bayesian statistical inference and two kinds of variable selection could be adopted for the model of total wealth, especially the Merton regression model; applying the hierarchical mixtures-of-experts method to combine different option pricing models, the model-combination is interpreted as the combination of several different assessments of the investor on how the target stock will develop over time, which is reformulated as combing several BSDE described the corresponding replicating portfolios,and we will show that combining several BS models with different volatility parameters will lead to an improved out-of-sample performance and by applying the mixture, new method can describe how much each of the candidate models is preferred according different categories of the moneyness and the term to expiration. Plus, when a constraint is appended to a classical regression model, new problems arise naturally, including how to build an constraint-dependent model, how to realize its identifiability of the new model, and how to construct an estimation. To solve these fundamental problems, we introduce a remodeling method to treat the variable constraint as a quasi instrumental-variable and further to correct the bias and identify the model.

英文关键词： Forward-Backward Stochastic Differential Equation;Bayesian Inference;Variable Selection;Hierarchical Mixtures-of-experts;Instrumental Variable