Research Article
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Stochastic integration with respect to a cylindrical special semi-martingale

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

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
Year 2022, Volume: 71 Issue: 4, 899 - 906, 30.12.2022
https://doi.org/10.31801/cfsuasmas.981876

Abstract

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
There are 19 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Saeed Hashemi Sababe 0000-0003-1167-5006

Publication Date December 30, 2022
Submission Date August 16, 2021
Acceptance Date April 22, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Hashemi Sababe, S. (2022). Stochastic integration with respect to a cylindrical special semi-martingale. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 899-906. https://doi.org/10.31801/cfsuasmas.981876
AMA Hashemi Sababe S. Stochastic integration with respect to a cylindrical special semi-martingale. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):899-906. doi:10.31801/cfsuasmas.981876
Chicago Hashemi Sababe, Saeed. “Stochastic Integration With Respect to a Cylindrical Special Semi-Martingale”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 899-906. https://doi.org/10.31801/cfsuasmas.981876.
EndNote Hashemi Sababe S (December 1, 2022) Stochastic integration with respect to a cylindrical special semi-martingale. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 899–906.
IEEE S. Hashemi Sababe, “Stochastic integration with respect to a cylindrical special semi-martingale”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 899–906, 2022, doi: 10.31801/cfsuasmas.981876.
ISNAD Hashemi Sababe, Saeed. “Stochastic Integration With Respect to a Cylindrical Special Semi-Martingale”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December 2022), 899-906. https://doi.org/10.31801/cfsuasmas.981876.
JAMA Hashemi Sababe S. Stochastic integration with respect to a cylindrical special semi-martingale. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:899–906.
MLA Hashemi Sababe, Saeed. “Stochastic Integration With Respect to a Cylindrical Special Semi-Martingale”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 899-06, doi:10.31801/cfsuasmas.981876.
Vancouver Hashemi Sababe S. Stochastic integration with respect to a cylindrical special semi-martingale. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):899-906.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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