Theory of Probability and Mathematical Statistics
Integral equations with respect to a general stochastic measure
V. M. Radchenko
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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.
Keywords: Stochastic measure, stochastic integral, stochastic differential equation, Besov space
Bibliography: 1. S. Ogawa, Stochastic integral equations for the random fields, Seminaire de Probabilites XXV, Springer, Berlin-Heidelberg, 1991, pp. 324-329.
2. T. Mikosch and R. Norvaiša, Stochastic integral equations without probability, Bernoulli 6 (2000), no. 3, 401-434.
3. V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231-251.
4. 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.
5. S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992.
6. 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.
7. G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman and Hall, London, 1994.
8. V. N. Radchenko, Integrals with respect to general random measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine 27 (1999). (Russian)
9. G. Curbera and O. Delgado, Optimal domains for L0-valued operators via stochastic measures, Positivity 11 (2007), no. 3, 399-416.
10. V. Radchenko, Besov regularity of stochastic measures, Statist. Probab. Lett. 77 (2007), no. 8, 822-825.
11. A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63-77.
12. 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.