Generalized Random Measures on Topological Spaces
Al-Obaidi, Ali Hussein Mahmood
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Our work deals with classes of random measures on σ-compact Hausdorff spaces perturbed by stochastic processes. We render a rigorous construction of the stochastic integral of functions of two variables and show that such an integral is a random measure. We establish a new Campbell-type formula that, along with a rigorous construction of modulation, leads to the intensity of a modulated random measure. We further introduce and study a marked Poisson random measure on a σ- compact Hausdorff space. The underlying parameters of this measure are changing in accordance with the evolution of a stochastic process. This generalized random measure has properties resembling those of the conventional Poisson random measure. We obtain an explicit formula for the probability distribution of such measure in the form of the Fourier- Stieltjes functional, show other notable properties including continuity in probability and quasi-independent increments, and discuss various applications of the generalized Poisson measure (modulated by a semi- Markov process) to astrophysics and finance.