Abstract
Stochastic integrals of real-valued functions with respect to general stochastic measures are considered in the chapter. For the integrator we assume the σ-additivity in probability only. The chapter contains a review of recent results concerning Besov regularity of stochastic measures, continuity of paths of stochastic integrals, and solutions of stochastic partial differential equations (SPDEs) driven by stochastic measure. Some important properties of stochastic integrals are proved. The Riemann-type integral of random function with respect to the Jordan content is introduced. For the heat equation in \(\mathbb{R}\), we consider the existence, uniqueness, and Hölder regularity of the mild solution. For a general parabolic SPDE in \({\mathbb{R}}^{d}\), we obtain the weak solution. Integrals of random functions with respect to deterministic measures in the equations are understood in Riemann sense.
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Acknowledgements
This research was partially supported by Alexander von Humboldt Foundation, grant no. UKR/1074615.
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Radchenko, V. (2014). Stochastic Partial Differential Equations Driven by General Stochastic Measures. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_9
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DOI: https://doi.org/10.1007/978-3-319-03512-3_9
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