Sankhya A

, Volume 80, Issue 2, pp 267–279 | Cite as

Direct Inversion Formulas for the Natural SFT

  • Shigeyoshi Ogawa


The stochastic Fourier transform, or SFT for short, is an application that transforms a square integrable random function f(t, ω) to a random function defined by the following series; \({\mathcal T}_{\epsilon , \varphi }f(t,\o ):= {\sum }_{n} \epsilon _{n} \hat {f}_{n}(\o )\varphi _{n}(t)\) where {𝜖n} is an 2-sequence such that 𝜖n ≠  0, ∀n and \(\hat {f}_{n}\) is the SFC (short for “stochastic Fourier coefficient”) defined by \(\hat {f}_{n}(\o )={{\int }_{0}^{1}} f(t,\o )\overline {\varphi _{n}(t)}dW_{t}\), a stochastic integral with respect to Brownian motion Wt. We have been concerned with the question of invertibility of the SFT and shown affirmative answers with concrete schemes for the inversion. In the present note we aim to study the case of a special SFT called “natural SFT” and show some of its basic properties. This is a follow-up of the preceding article (Ogawa,S.,“A direct inversion formula for SFT”, Sankhya-A 77-1 (2015)).

Keywords and phrases

Brownian motion Stochastic integrals Fourier series 

AMS (2000) subject classification.

Primary: 60H05, 60H07 Secondary: 42A61 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ogawa, S. (1979). Sur le produit direct du bruit blanc par lûi-même., Comptes Rendus Acad Sci, Paris t.288, Série A, 359–362.Google Scholar
  2. Ogawa, S. (1984). Une remarque sur l’approximation de l’integrale stochastique du type noncausal par une suite des integrales de Stieltjes. Tôhoku Math. J. 36, 1, 41–48. Tôhoku Univ.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ogawa, S. (1986). On the stochastic integral equations of Fredholm type. “Waves and Patterns” (monograph), Kinokuniya and North-Holland. pp. 597–605.Google Scholar
  4. Ogawa, S. (1991). On a stochastic integral equation for the random fields. S.éminaire de Proba. 25, 324–339. Springer.MathSciNetzbMATHGoogle Scholar
  5. Ogawa, S. (2008). Real time scheme for the volatility estimation in the presence of microstructure noise. Monte Carlo Methods and Applications 14, 4, 331–342.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ogawa, S. (2013). Stochastic Fourier transformation. Stochastics 85, 2, 286–294.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ogawa, S. and Uemura, H. (2014). Stochastic Fourier coefficient – case of noncausal functions. J. Theoret. Probab. 27, 2, 370–382.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ogawa, S and Uemura, H. (2014). Identification of noncausal Itô processes from the stochastic Fourier coefficients. Bull. Sci. Math. 138, 1, 147–163.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Ogawa, S. (2015). A direct inversion formula for the SFT. Sankhya A 77, 1, 30–45.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ogawa, S. (2016). BPE and a noncausal Girsanov’s theorem, Sankhya A.

Copyright information

© Indian Statistical Institute 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRitsumeikan UniversityShigaJapan

Personalised recommendations