Inhomogeneous Random Evolutions and their Applications in Finance

在物理语言中，随机进化是一个动态系统在随机环境中的模型，它的状态方程是随机变化的；例如，在不同波动率之间转换的股票/资产。在数学语言中，随机进化是由一些随机过程驱动的半群算子（描述系统的进化）的产物。随机过程定义了随机演化的名称：马尔可夫、半马尔可夫等。同样，根据进化的结构，我们有连续的、离散的、均匀的、非均匀的进化等等。欧几里得空间中的马尔可夫随机演化在文献中通常称为隐马尔可夫模型或状态切换模型。随机进化在20世纪70年代开始被研究，因为它们在金融、保险、生物学、存储、排队和风险理论等方面有潜在的应用。在这个提议中，我们引入了一类新的随机进化，即非齐次（或时间非齐次，不具有平稳增量）半马尔可夫随机进化，并考虑了它们在金融、能源和环境金融中的应用。***该建议的新颖之处在于三个方面：i)研究了一种新的非齐次半马尔可夫过程（其中不同环境状态之间的过渡时间分布是任意的，而不是像马尔可夫情况那样具体），它改变了进化的体制；Ii)研究了一组描述系统演化的非齐次半群算子所构造的新的非齐次随机演化；（三）在金融、能源和环境金融领域的应用。据我们所知，只研究了齐次随机演化（基于齐次马尔可夫或半马尔可夫过程，以及齐次半算子群）及其应用。随着金融业的全面电脑化，记录数据的数量，如高频时间依赖性数据，呈爆炸式增长。因此，在非齐次或时间相关的半马尔可夫设置中研究这些数据似乎是很自然的。此外，天气衍生品的应用也可用于对冲能源衍生品和构建对冲投资组合。天气衍生品也可用于环境案例，例如与加拿大农业部门的生物技术一起使用，以减少作物对环境变化的脆弱性，并减少与气候变化影响相关的风险。本研究计划将是一个综合的工具，不仅融入随机过程的一般理论，而且融入其在定量金融，风险管理，能源和环境金融领域的应用。我们的模型和方法将应用于加拿大的金融、能源和环境市场，从而增加我们对加拿大环境和可持续发展问题的认识和意识。

A random evolution, in physical language, is a model for a dynamical system in random environment, whose equation of state is subject to random variations; for example, a stock/asset that switches between different volatilities. In mathematical language, a random evolution is a product of semi-group of operators (describing the evolution of the system) driven by some stochastic process. The stochastic process defines the name for the random evolutions: Markov, semi-Markov, etc. Also, depending on structure of the evolution, we have continuous, discrete, homogeneous, inhomogeneous evolutions, etc. Markov random evolutions in Euclidean spaces are usually called in the literature hidden Markov or regime-switching models. Random evolutions began to be studied in the 1970's, because of their potential applications in finance, insurance, biology, storage, queuing and risk theories, to name a few. In this proposal we introduce a new class of random evolutions, namely, inhomogeneous (or time-inhomogeneous, which does not possess stationary increments) semi-Markov random evolutions and consider their applications in finance, energy and environmental finance. ***The novelty of the proposal is threefold: i) study of a new inhomogeneous semi-Markov process (where transition time distribution between different states of environment is arbitrary, not specific as in Markov case), which switches regimes of evolution; ii) study of a new inhomogeneous random evolutions, constructed by a family of inhomogeneous semi-group of operators, which describe the evolution of our system; iii) applications in finance, energy and environmental finance. To our best knowledge, only homogeneous random evolutions (based on homogeneous Markov or semi-Markov processes, and on homogeneous semi-group of operators) and their applications have been studied.***With financial industry becoming fully computerized, the amount of recorded data, such as high-frequency time-depended data, has exploded. It is seemed then natural to study these data in an inhomogeneous or time-dependent semi-Markov setting. Also, such application as weather derivatives can be useful to hedge the energy derivatives and to construct the hedging portfolios.  The weather derivatives can also be used in environmental case, e.g., together with biotechnology in Canada agriculture sector to reduce vulnerability of crops to environmental changes, and to reduce the risk associated with climate change impact. This research proposal will be a comprehensive tool to integrate not only into the general theory of stochastic processes, but also into the field of their applications in quantitative finance, risk management, and energy and environmental finance. Our models and methods will be applied to Canadian finance, energy and environmental markets, and thus increase our knowledge and awareness of Canadian environmental and sustainability issues.