Abstract
We are concerned with the question whether and how a random function f(t,ω) can be uniquely determined by its image of SFT (i.e. stochastic Fourier transformation). The question was first posed by the author in the study of various problems of stochastic analysis (Ogawa 1979, 1985) and has been studied again recently in the papers Ogawa (1986), Ogawa and Uemura (2013, 2014) where some affirmative answers to the question as well as the schemes for inversion of SFT are given. In these papers the problem has been studied in the framework of Homogeneous Chaos which we feel clumsy partly because there all statements and schemes for the inversion are expressed in terms of the infinite sequence of representing kernels of the given Wiener functional and partly because for the execution of the inversion scheme developed there we need complete data of the underlying Brownian motion. The aim of the present note is to show an elementary approach to the problem, that does not rely on such heavy frameworks like Homogeneous Chaos, and give a direct formula for the inversion of the SFT.
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References
Itô K. (1951) Multiple Wiener integrals. J. Math. Soc. Jpn. 3(1):157–169
Malliavin P., Mancino M. (2002) Fourier series method for measurement of multivariate volatilities. Finance Stoch. 6-1:49–61
Malliavin P., Thalmeyer A. (2009) Stochastic Calculus of Variations in Mathematical Finance. Springer
Ogawa S. (1970a) On a Riemann definition of the stochastic integral, (I). Proc. Jpn. Acad. 46:153–157
Ogawa S. (1970b) On a Riemann definition of the stochastic integral, (II). Proc. Jpn. Acad. 46:158–161
Ogawa S. (1979) Sur le produit direct du bruit blanc par lui-même. C. R. Acad. Sci., t.288 Sér. A, 359–362
Ogawa S. (1984) Quelques propriétés de l’integrale stochastic du type noncausal. Jpn. J. Appl. Math. Kinokuniya 1:405–416
Ogawa S. (1985) The stochastic integral of noncausal type as an extension of the symmetric integrals. Jpn. J. Appl. Math. 2(1):229–240
Ogawa S. (1986) On the stochastic integral equation of Fredholm type. In Patterns and Waves (monograph). Studies in Math and Its Appl., (Kinokuniya) vol. 18, pp. 597–606
Ogawa S. (1991) On a stochastic integral equation for the random fields. In Séminaire de Proba vol. 25. Springer, pp. 324–339
Ogawa S. (2008) Real time scheme for the volatility estimation in the presence of microstructure noise. Monte Carlo Methods Appl. 14(4):331–342
Ogawa S. (2013) Stochastic Fourier transformation. Stochastics 85-2:286–294
Ogawa S., Uemura H. (2013) On a stochastic Fourier coefficient: case of noncausal functions. J. Theoret. Probab.
Ogawa S., Uemura H. (2014) Identification of a noncausal Itô process from the stochastic Fourier coefficients. Bull. Sci. Math.. http://dx.doi.org/10.1016/j.bulsci.2013.12.003
Seveljakov A.Ju. (1978) Stochastic differentiation and integration of functionals of a Wiener process. Theor. Slucainyh Processov, vyp 6:123–131
Skorokhod A.V. (1975) On a generalization of a stochastic integral. Theory Probab. Appl. 20:219–233
Wiener N. (1938) The homogeneous chaos. Amer. J. Math., LV, 4
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Ogawa, S. A Direct Inversion Formula for SFT. Sankhya A 77, 30–45 (2015). https://doi.org/10.1007/s13171-014-0056-1
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DOI: https://doi.org/10.1007/s13171-014-0056-1