Stochastic Partial Differential Equations Driven by General Stochastic Measures

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 90)

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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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisTaras Shevchenko National University of KyivKyivUkraine

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