英文标题：
《On spatially irregular ordinary differential equations and a pathwise
  volatility modelling framework》
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作者：
Ryan McCrickerd
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最新提交年份：
2021
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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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中文摘要：
本文开发了一个新的金融价格过程建模框架，如股票价格或外汇汇率。这可以通过价格平方波动率的时间积分或“累积方差”与基于Ito演算的传统框架相关联。在新框架中，相应的过程严格递增，求解随机常微分方程（ODE），并由几何布朗运动组成。新框架不依赖于随机演算，因此可以使用无概率ODE技术和泛函分析在路径基础上研究过程。考虑的ODE依赖于“空间不规则”的连续驱动函数，这意味着它们不需要具有任何空间规则性属性，例如Holder连续性。然而，它们在时间上严格地增加，因此在时间上是不对称的。当选择合理的初值时，初值问题（IVP）的解也严格递增，并且这类IVP的解集包含非负实上的所有可微双射。这使得能够通过解的时间导数对任何非负波动路径进行建模，该路径在时间间隔内不为零。尽管存在这种普遍性，但新的适定性结果确定了解决方案在时间上的唯一性。探索该框架的动机来自于它与一个随时间变化的赫斯顿波动率模型的联系。该框架展示了赫斯顿价格过程如何收敛到NIG征税过程的推广，并揭示了综合CIR过程和IG过程之间的更深层次的关系。在此框架内，定义了一个“Riemann-Liouville-Heston”鞅模型，该模型将这些关系推广到分数阶鞅模型。对该模型的隐含波动率进行了模拟，显示了领先波动率模型的特征。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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