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Stochastic Partial Differential Equations Driven by General Stochastic Measures

In: Modern Stochastics and Applications

Author

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  • Vadym Radchenko

    (Taras Shevchenko National University of Kyiv)

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.

Suggested Citation

  • Vadym Radchenko, 2014. "Stochastic Partial Differential Equations Driven by General Stochastic Measures," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 143-156, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_9
    DOI: 10.1007/978-3-319-03512-3_9
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    Cited by:

    1. Radchenko, Vadym, 2019. "Averaging principle for the heat equation driven by a general stochastic measure," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 224-230.

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