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

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.

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。

在金融和经济学等领域，我们关心包含变量约束和时间相依数据的回归问题。为此，本项目引入重构方法把变量约束处理成拟工具变量，并且进一步修正偏误以及识别模型，使用轮廓估计的方法估计新模型中的非参数回归函数和参数，得到了估计量的相合性和渐近正态性。通过模拟研究了小样本性质，并用真实的股票期权数据验证了该模型。在高维复杂数据分析方面，利用多源组学数据研究致病基因预测方法和链路预测方法，发现致病基因与必需基因的关联较少，给出了网络间相依性分析的方法与冷启动链路预测的方法。此外，研究了广义矩估计的模型平均方法，在广义矩估计条件下提出了通过最小化目标参数平均估计量的渐近方差获取权重，可有效提升估计的稳健性。相关成果已有5篇发表或接收，其中SCI论文3篇，核心论文2篇。本项目的研究扩展了随机分析领域FBSDE模型的统计推断、复杂动态网络数据的统计推断，对金融产品定价、风险控制和生物信息学有指导作用。