Inhomogeneous Random Evolutions and their Applications in Finance

在物理语言中，随机演化是随机环境中动态系统的模型，其状态方程受到随机变化的影响；例如，在不同波动性之间切换的股票/资产。在数学语言中，随机演化是由某个随机过程驱动的半群算子(描述系统的演化)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。根据演化的结构，我们也有连续的、离散的、齐次的、非齐次的演化等。文献中通常将欧氏空间中的马尔可夫随机演化称为隐马尔可夫模型或制度转换模型。随机演化的研究始于20世纪70年代的S，因为它在金融、保险、生物学、存储、排队和风险理论等领域具有潜在的应用价值。在这个方案中，我们引入了一类新的随机演化，即非齐次(或时间非齐次，不具有平稳增量)半马尔可夫随机演化，并考虑了它们在金融、能源和环境金融中的应用。*该建议的新颖性有三个方面：i)研究一个新的非齐次半马尔可夫过程(其中环境不同状态之间的转移时间分布是任意的，而不是像马尔可夫情形那样是特定的)，它改变了进化机制；ii)研究由一族非齐次半群算子构造的新的非齐次随机演化，它描述了我们系统的演化；iii)在金融、能源和环境金融中的应用。据我们所知，只有齐次随机演化(基于齐次马尔科夫过程或半马尔可夫过程，以及齐次半马尔可夫算子群)及其应用被研究过。*随着金融业的完全计算机化，记录的数据量呈爆炸式增长，例如高频的时间相关数据。当时，在不均匀或依赖时间的半马尔可夫环境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以用于对冲能源衍生品和构建套期保值投资组合。天气衍生品也可以用于环境情况，例如与加拿大农业部门的生物技术一起使用，以降低作物对环境变化的脆弱性，并降低与气候变化影响相关的风险。这项研究提案将是一个全面的工具，不仅可以整合到随机过程的一般理论中，而且还可以整合到它们在量化金融、风险管理以及能源和环境金融中的应用领域。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认识。

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.