论空间非正则常微分方程与路径式波动率建模框架
On spatially irregular ordinary differential equations and a pathwise volatility modelling framework
AI总结：
本文提出不依赖随机微积分的路径式波动率建模框架，基于空间非正则常微分方程实现波动率路径建模，确立正向解唯一性，关联时变Heston模型并构建分数阶鞅模型，模拟验证其具备主流波动率模型特征。
摘要：
本文构建了一个用于金融价格过程（如股票价格或外汇汇率）建模的新框架。该框架可通过价格平方波动率的时间积分（即“累积方差”）与传统基于伊藤微积分的框架建立关联。在新框架中，对应过程具有严格递增性，满足随机常微分方程（ODE），并与几何布朗运动复合。该框架不依赖随机微积分，因此可采用无概率的常微分方程技术与泛函分析，在路径层面研究相关过程。\n本文所研究的常微分方程依赖于“空间非正则”的连续驱动函数，即这类函数无需具备赫尔德连续性等任何空间正则性性质，但它们在时间维度上严格递增，因此具有时间不对称性。当选取合理的初始值时，初值问题（IVP）的解同样严格递增，且已证明这类初值问题的解集包含非负实数域上的所有可微双射。这使得我们能够通过解的时间导数，对任意在区间上不为零的非负波动率路径进行建模。尽管具备这种一般性，新的适定性结果仍确立了时间正向解的唯一性。\n探索该框架的动机源于它与时变Heston波动率模型的关联。该框架揭示了Heston价格过程如何收敛于NIG Lévy过程的推广形式，并展现了积分CIR过程与IG过程之间更深层的联系。在此框架内，本文定义了一种“Riemann-Liouville-Heston”鞅模型，将上述关系推广至分数阶对应情形。本文对该模型的隐含波动率进行了模拟，结果显示其具备主流波动率模型的特征性表现。
英文摘要：
This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's squared volatility, or `cumulative variance'. In the new framework, corresponding processes are strictly increasing, solve random ordinary differential equations (ODEs), and are composed with geometric Brownian motion. The new framework has no dependence on stochastic calculus, so processes can be studied on a pathwise basis using probability-free ODE techniques and functional analysis. The ODEs considered depend on continuous driving functions which are `spatially irregular', meaning they need not have any spatial regularity properties such as Holder continuity. They are however strictly increasing in time, thus temporally asymmetric. When sensible initial values are chosen, initial value problem (IVP) solutions are also strictly increasing, and the solution set of such IVPs is shown to contain all differentiable bijections on the non-negative reals. This enables the modelling of any non-negative volatility path which is not zero over intervals, via the time derivative of solutions. Despite this generality, new well-posedness results establish the uniqueness of solutions going forwards in time. Motivation to explore this framework comes from its connection with a time-changed Heston volatility model. The framework shows how Heston price processes can converge to a generalisation of the NIG Levy process, and reveals a deeper relationship between integrated CIR processes and the IG process. Within this framework, a `Riemann-Liouville-Heston' martingale model is defined which generalises these relationships to fractional counterparts. This model's implied volatilities are simulated, and exhibit features characteristic of leading volatility models.
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