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.