Journal of Theoretical Probability

, Volume 32, Issue 4, pp 1973–1989 | Cite as

On the Identification of Noncausal Wiener Functionals from the Stochastic Fourier Coefficients

  • Kiyoiki Hoshino
  • Tetsuya KazumiEmail author


Let \((B_t)_{t\in [0,1]}\) be a one-dimensional Brownian motion starting at the origin and \((e_n)_{n\in \mathbb {N}}\) a complete orthonormal system of \(L^2([0,1],\mathbb {C})\). Ogawa and Uemura (J Theor Probab 27:370–382, 2014) defined the stochastic Fourier coefficient \(\hat{a}_{n}(\omega )\) of a random function \(a(t,\omega )\) by \(\hat{a}_{n}(\omega ):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t\), where \(\int \,\delta B\) stands for the Skorokhod integral, and considered the question of whether the random function \(a(t,\omega )\) can be identified from the totality of \(\hat{a}_{n}\), \(n\in \mathbb {N}\). Ogawa and Uemura (Bull Sci Math 138:147–163, 2014) discussed the identification problem for the stochastic Fourier coefficient \(\mathcal {F}_{n}(\delta X)\) of a Skorokhod-type stochastic differential \(\delta X_t=a(t,\omega )\,\delta B_t+b(t,\omega )\,\hbox {d}t\), where \(\mathcal {F}_{n}(\delta X)\) is defined by \(\mathcal {F}_{n}(\delta X):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t +\int _0^1 \overline{e_n(t)} b(t,\omega ) \, \hbox {d}t\). They obtained affirmative answers to these questions under certain conditions. In this paper, we ask similar questions and we obtain affirmative answers under weaker conditions. Moreover, we consider the identification problem for the stochastic Fourier coefficients based on Ogawa integral.


Stochastic Fourier coefficient Skorokhod integral Ogawa integral Multiple Wiener–Itô integral Brownian motion Noncausal Wiener functional 

Mathematics Subject Classification (2010)

Primary 60H05 Secondary 60H07 



The authors would like to express their gratitude to Professor Shigeyoshi Ogawa and Professor Hideaki Uemura for discussions and comments about the subject on various occasions. They would also like to thank the anonymous referees for their helpful comments.


  1. 1.
    Hoshino, K., Kazumi, T.: On the Ogawa integrability of noncausal Wiener functionals, preprint (2017)Google Scholar
  2. 2.
    Itõ, K.: Stochastic integral. Proc. Imp. Acad. Tokyo 20, 519–524 (1944)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Itõ, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ogawa, S.: Sur le produit direct du bruit blanc par lui-même. C. R. Acad. Sc. Paris Sér. A 288, 359–362 (1979)zbMATHGoogle Scholar
  5. 5.
    Ogawa, S.: Quelques propriétés de l’intégrale stochastique du type noncausal. Jpn. J. Appl. Math. 1, 405–416 (1984)CrossRefGoogle Scholar
  6. 6.
    Ogawa, S.: Une remarque sur l’approximation de l’intégrale stochastique du type noncausal par une suite des intégrales de Stieltjes. Tohoku Math. J. (2) 36(1), 41–48 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ogawa, S.: The stochastic integral of noncausal type as an extension of the symmetric integrals. Jpn. J. Appl. Math. 2, 229–240 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ogawa, S.: On the stochastic integral equation of Fredholm type. In: Patterns and Waves (monograph). Stud. Math. Appl. Kinokuniya 18, 597–606 (1986)Google Scholar
  9. 9.
    Ogawa, S.: Stochastic integral equations for the random fields. Semin. Probl. XXV, 324–339 (1991)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ogawa, S.: On a stochastic Fourier transformation. Stochastics 85, 286–294 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ogawa, S.: A direct inversion formula for the SFT. Sankhya A 77, 30–45 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ogawa, S., Uemura, H.: On a stochastic Fourier coefficient: case of noncausal functions. J. Theor. Probab. 27, 370–382 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ogawa, S., Uemura, H.: Identification of a noncausal Itõ process from the stochastic Fourier coefficients. Bull. Sci. Math. 138, 147–163 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ogawa, S., Uemura, H.: On the identification of noncausal functions from the SFCs. RIMS Kôkyûroku 1952, 128–134 (2015)Google Scholar
  15. 15.
    Skorokhod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1975)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of ScienceOsaka Prefecture UniversitySakaiJapan

Personalised recommendations