Integral equations with respect to a general stochastic measure
Author:
V. M. Radchenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 91 (2015), 169-179
MSC (2010):
Primary 60H20, 60H05, 60G57
DOI:
https://doi.org/10.1090/tpms/975
Published electronically:
February 4, 2016
MathSciNet review:
3364132
Full-text PDF Free Access
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Additional Information
Abstract: An integral with respect to a general stochastic measure is defined for random functions whose trajectories belong to a Besov space. The existence and uniqueness of solutions of some stochastic equations involving such integrals are established.
References
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- Thomas Mikosch and Rimas Norvaiša, Stochastic integral equations without probability, Bernoulli 6 (2000), no. 3, 401–434. MR 1762553, DOI https://doi.org/10.2307/3318668
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- V. N. Radchenko, Integrals with respect to general random measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine 27 (1999). (Russian)
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- Anna Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63–77. MR 1750304
- V. N. Radchenko, On a definition of the integral of a random function, Teor. Veroyatnost. i Primenen. 41 (1996), no. 3, 677–682 (Russian, with Russian summary); English transl., Theory Probab. Appl. 41 (1996), no. 3, 597–601 (1997). MR 1450086
References
- S. Ogawa, Stochastic integral equations for the random fields, Seminaire de Probabilites XXV, Springer, Berlin–Heidelberg, 1991, pp. 324–329. MR 1187790 (94b:60073)
- T. Mikosch and R. Norvaiša, Stochastic integral equations without probability, Bernoulli 6 (2000), no. 3, 401–434. MR 1762553 (2001h:60100)
- V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554 (2010j:60157)
- V. Radchenko, Stochastic partial differential equations driven by general stochastic measures, Modern Stochastics and Applications (V. Korolyuk, N. Limnios, Yu. Mishura, L. Sakhno, and G. Shevchenko, eds.), Springer/Cham Heidelberg, 2014, pp. 143–156. MR 3236073
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- J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 27 (2001), no. 2, 197–206. MR 1822771 (2002b:60096)
- G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman and Hall, London, 1994. MR 1280932 (95f:60024)
- V. N. Radchenko, Integrals with respect to general random measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine 27 (1999). (Russian)
- G. Curbera and O. Delgado, Optimal domains for ${L}^0$-valued operators via stochastic measures, Positivity 11 (2007), no. 3, 399–416. MR 2336205 (2008g:46063)
- V. Radchenko, Besov regularity of stochastic measures, Statist. Probab. Lett. 77 (2007), no. 8, 822–825. MR 2369688 (2009c:60126)
- A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63–77. MR 1750304 (2001e:46058)
- V. N. Radchenko, On a definition of the integral of a random function, Teor. Veroyatnost. Primenen. 41 (1996), no. 3, 677–682; English transl. in Theory Probab. Appl. 41 (1997), no. 3, 597–601. MR 1450086 (98f:60002)
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Additional Information
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, National Taras Shevchenko University, Kyiv 01601, Ukraine
Email:
vradchenko@univ.kiev.ua
Keywords:
Stochastic measure,
stochastic integral,
stochastic differential equation,
Besov space
Received by editor(s):
August 29, 2014
Published electronically:
February 4, 2016
Article copyright:
© Copyright 2016
American Mathematical Society