Korean J. Math.  Vol 29, No 3 (2021)  pp.639-647
DOI: https://doi.org/10.11568/kjm.2021.29.3.639

Reproducing kernel Hilbert space based on special integrable semimartingales and stochastic integration

Saeed Hashemi Sababe, Maryam Yazdi, Mohammad Mehdi Shabani

Abstract


In this paper, we consider the integral of a stochastic process with respect of a sequence of  square integrable semimartingales. By this integrals, we construct a reproducing kernel Hilbert space and study the correspondence between this space with the concepts of arbitrage and viability in mathematical finance.


Keywords


stochastic integration, continuous semimartingales, reproducing kernels

Subject classification

60H05; 60B11; 60G44; 47D06

Sponsor(s)



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References


M. Al-Smadi, Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation, Ain Shams Eng J. 9 (2018), 2517–2525. (Google Scholar)

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-–404. (Google Scholar)

A. Berlinet and C. Tomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Springer Science, Business, Media New York, 2004. (Google Scholar)

T. Choulli and C. Stricker, Deux application de la d ́ecomposotion de Galtchouk-Kunita-Watanabe, S ́eminaire De Probabilit ́es. 30 (1996), 12-–23. (Google Scholar)

T. Choulli, J. Deng and J. Ma, How non-arbitrage, viability and num ́eraire portfolio are related., Finance Stoch. 19 (4) (2015), 719–741. (Google Scholar)

C. Cuchiero, I. Klein and J. Teichmann, A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting., Theory Probab. Appl. 65(3) (2020), 388–404; translation from Teor. Veroyatn. Primen. 65 (3) (2020), 498–520. (Google Scholar)

C. Cuchiero, I. Klein and J. Teichmann, A new perspective on the fundamental theorem of asset pricing for large financial markets. , Theory Probab. Appl. 60(4) (2016), 561–579; translation from Teor. Veroyatn. Primen. 60 (4) (2015), 660–685 . (Google Scholar)

F. Delbaen and H. Shirakawa, A note on the no arbitrage condition for international financial markets. , Financ. Eng. Jpn. Mark. 3 (3) (1994), 239–251. (Google Scholar)

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing. , Math. Ann. 300, 33 (3) (1994), 463–520. (Google Scholar)

F. Delbaen and W. Schachermayer, Applications to mathematical finance., Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier. 367–391 (2001). (Google Scholar)

F. Delbaen and W. Schachermayer, Arbitrage possibilities in Bessel processes and their relations to local martingales., Probab. Theory Relat. Fields, 102 (3) (1995), 357–366. (Google Scholar)

F. Delbaen and W. Schachermayer, A simple counterexample to several problems in the theory of asset pricing. , Math. Finance. 8 (1) (1998), 1–11. (Google Scholar)

F. Delbaen and W. Schachermayer, The Banach space of workable contingent claims in arbitrage theory. , Ann. Inst. Henri Poincar ́e, Probab. Stat. 33 (1) (1997), 113–144. (Google Scholar)

F. Delbaen and W. Schachermayer, The mathematics of arbitrage. , Springer Finance. Springer-Verlag, Berlin, 2006. (Google Scholar)

C. Dellacherie, Quelques applications du lemme de Borel–Cantelli `a la th ́eeorie des semimartin-gales, S ́eminaire de Probabilit ́es XII, Lecture Notes in Math., 649, 742-745, Springer, 1978. (Google Scholar)

M. De Donno, P. Guasoni and M. Pratelli, Super-replication and utility maximization in large financial markets. , Stochastic Processes Appl. 115 (12) (2005), 2006–2022. (Google Scholar)

S. Hashemi Sababe and A. Ebadian, Some properties of reproducing Kernel Banach and Hilbert spaces, Sahand commun. math. anal., 12 (1) (2018), 167–177. (Google Scholar)

S. Hashemi Sababe, A. Ebadian and Sh. Najafzadeh, On reproducing property and 2-cocycles, Tamkang J. Math., 49 (2) (2018), 143–153. (Google Scholar)

S. Hashemi Sababe, 0 On 2-inner product and reproducing property, Korean J. Math., 28 (4) (2020), 973–984. (Google Scholar)

J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes, Springer-Verlag Berlin Heidelberg, (1987) (Google Scholar)

Y.M. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer.,In Statistics and control of stochastic processes (Moscow,1995/1996), pages 191–203. World Sci. Publ., River Edge, NJ, 1997. (Google Scholar)

C. Kardaras, Stochastic integration with respect to arbitrary collections of continuous semi-martingales and applications to Mathematical Finance, arXiv:1908.03946v2 [math.PR], 2019. (Google Scholar)

H. Kunita and Sh. Watanabe On square integrable martingales, Nagoya Math. J. 30 (1967), 209–245. https://projecteuclid.org/euclid.nmj/1118796812 (Google Scholar)

R. Mikulevicius and B.L. Rozovskii, Normalized stochastic integrals in topological vector spaces, S ́eminaire de Probabilit ́es XXXII, Lecture Notes in Math., Springer, 137-165, Springer, 1998. (Google Scholar)

R. Mikulevicius and B.L. Rozovskii, Martingale problems for stochastic PDE’s, Amer. Math. Soc., 64 (1999), 243–326. (Google Scholar)

J. M ́emin, Espaces de semi martingales et changement de probabilit ́e, Z Wahrscheinlichkeit 52 (1980), 9–39. (Google Scholar)

M. Schuld and N. Killoran, Quantum Machine Learning in Feature Hilbert Spaces, Phys Rev Lett, 122 (2018), 040504(1)-040504(6). (Google Scholar)

S. Vahdati, M. Fardi and M. Ghasemi, Option pricing using a computational method based on reproducing kernel, J Comput Appl Math, 328 (2018), 252–266. (Google Scholar)


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