Inhomogeneous Random Evolutions and their Applications in Finance

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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.