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Theory of Probability and Mathematical Statistics

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Mild solution of the parabolic equation driven by a $ \sigma$-finite stochastic measure


Authors: O. O. Vertsimakha and V. M. Radchenko
Translated by: N. N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 97 (2017).
Journal: Theor. Probability and Math. Statist. 97 (2018), 17-32
MSC (2010): Primary 60H15, 60G57, 60G17
DOI: https://doi.org/10.1090/tpms/1045
Published electronically: February 21, 2019
MathSciNet review: 3745996
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Abstract: Stochastic parabolic equation driven by a $ \sigma $-finite stochastic measure in the interval $ [0,T]\times \mathbb{R}$ is studied. The only condition imposed on the stochastic integrator is its $ \sigma $-additivity in probability on bounded Borel sets. The existence, uniqueness, and Hölder continuity of a mild solution are proved. These results generalize those known earlier for usual stochastic measures.


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Additional Information

O. O. Vertsimakha
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: oksana.vertsim@gmail.com

V. M. Radchenko
Affiliation: Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: vradchenko@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/1045
Keywords: Stochastic measure, $\sigma$-finite {\SM}, stochastic parabolic equation, mild solution, H\"older continuity
Received by editor(s): May 15, 2017
Published electronically: February 21, 2019
Article copyright: © Copyright 2019 American Mathematical Society