Inhomogeneous Random Evolutions and their Applications in Finance

随机演化，在物理语言中，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，股票/资产在不同的波动率之间切换。在数学语言中，随机演化是由一些随机过程驱动的算子半群（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，等等。此外，根据演化的结构，我们有连续的，离散的，齐次的，非齐次的演化，等等。随机演化在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的潜在应用。在这个建议中，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，并考虑其在金融，能源和环境金融的应用。

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.