Stochastic integration with respect to a cylindrical special semi-martingale

Year 2022, Volume: 71 Issue: 4, 899 - 906, 30.12.2022

Saeed Hashemi Sababe

https://doi.org/10.31801/cfsuasmas.981876

Abstract

In this research, we introduce the stochastic integration with respect to a cylindrical special semi-martingale, which is a specific case of general integration, with specific properties of special semi-martingales.

Keywords

Cylindrical martingale , special martingales , stochastic integration , Banach space

References

• Brzezniak, Z., Van Neerven, J.M.A.M., Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem, Studia Math., 143(1) (2000), 43-74.
• Criens, D., Cylindrical martingale problems associated with Levy generators, J. Theoret. Probab., 32(3) (2019), 1306–1359. https://doi.org/10.48550/arXiv.1706.06049
• Di Girolami, C., Fabbri, G., Russo, F., The covariation for Banach space valued processes and applications, Metrika, 77(1) (2014), 51-104. https://doi.org/10.48550/arXiv.1301.5715
• Emery, M., Une Topologie Sur Lespace Des Semimartingales, Sem. Probab. XIII. Univ. Strasbourg, 260–280, Lecture Notes in Math. 721, Springer, 1979.
• Fonseca-Mora, C.A., Semimartingales on duals of nuclear spaces, Electron. J. Probab., 25(36) (2020). https://doi.org10.1214/20-EJP444
• Hashemi Sababe, S., Yazdi M., Shabani, M.M., Reproducing kernel Hilbert space based on special integrable semimartingales and stochastic integration, Korean J. Math., 29(3) (2021), 639–647. https://doi.org/10.11568/kjm.2021.29.3.639
• Jacod, J., Shiryaev, A.N., Limit Theorems for Stochastic Processes, Springer, 2003.
• Kalinichenko, A.A., An approach to stochastic integration in general separable Banach spaces, Potential Anal., 50(4) (2019), 591–608. https://doi.org/10.1007/s11118-018-9696-4
• Kalton, N.J., Weis, L.W.,The $H^{\infty}$-calculus and square function estimates, Nigel J. Kalton Selecta, 1 (2016), 715-764. https://doi.org/10.48550/arXiv.1411.0472
• Kardaras, C., On the closure in the Emery topology of semimartingale wealth-process sets, Ann. Appl. Probab., 23(4) (2013), 1355–1376. http://dx.doi.org/10.1214/12-AAP872
• Kumar, U., Riedle, M., The stochastic Cauchy problem driven by a cylindrical Levy process, Electron. J. Probab., 25(10), (2020). https://doi.org/10.48550/arXiv.1803.04365
• Memin, J., Espaces de semimartingales et changement de probabilite, Z. Wahrsch. Verw. Gebiete, 52(1) (1980), 9–39. https://doi.org/10.1007/BF00534184
• Metivier, M., Pellaumail, J., Stochastic Integration, Probability and Mathematical Statistics, Academic Press, 1980,
• Mnif, M., Pham, H., Stochastic optimization under constraints, Stochastic Process. Appl., 93 (2001), 149-180. https://doi.org/10.1016/S0304-4149(00)00089-2
• Ondrejat, M., Brownian representations of cylindrical local martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces, Czechoslovak Math. J., 55(130) (2005), 1003–1039. https://doi.org/10.1007/s10587-005-0084-z
• Rudin, W., Real and Complex Analysis, McGraw-Hill Book Co., 1987.
• Suchanecki, Z., Weron, A., Decomposability of cylindrical martingales and absolutely summing operators, Math. Z., 185(2) (1984), 271–280. https://doi.org/10.1007/BF01181698
• Sun, X., Xie, L., Xie, Y., Pathwise uniqueness for a class of SPDEs driven by cylindrical-stable processes, Potential Anal., 53(2) (2020), 659–675. https://doi.org/10.1007/s11118-019-09783-x
• Veraar, M., Yaroslavtsev, I., Cylindrical continuous martingales and stochastic integration in infinite dimensions, Electron. J. Probab., 21(59) (2016). https://doi.org/10.1214/16-EJP7