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

Kiyoiki Hoshino () and Tetsuya Kazumi ()
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Kiyoiki Hoshino: Osaka Prefecture University
Tetsuya Kazumi: Osaka Prefecture University

Journal of Theoretical Probability, 2019, vol. 32, issue 4, 1973-1989

Abstract: Abstract Let $$(B_t)_{t\in [0,1]}$$ ( B t ) t ∈ [ 0 , 1 ] be a one-dimensional Brownian motion starting at the origin and $$(e_n)_{n\in \mathbb {N}}$$ ( e n ) n ∈ N a complete orthonormal system of $$L^2([0,1],\mathbb {C})$$ L 2 ( [ 0 , 1 ] , C ) . Ogawa and Uemura (J Theor Probab 27:370–382, 2014) defined the stochastic Fourier coefficient $$\hat{a}_{n}(\omega )$$ a ^ n ( ω ) of a random function $$a(t,\omega )$$ a ( t , ω ) by $$\hat{a}_{n}(\omega ):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t$$ a ^ n ( ω ) : = ∫ 0 1 e n ( t ) ¯ a ( t , ω ) δ B t , where $$\int \,\delta B$$ ∫ δ B stands for the Skorokhod integral, and considered the question of whether the random function $$a(t,\omega )$$ a ( t , ω ) can be identified from the totality of $$\hat{a}_{n}$$ a ^ n , $$n\in \mathbb {N}$$ n ∈ N . Ogawa and Uemura (Bull Sci Math 138:147–163, 2014) discussed the identification problem for the stochastic Fourier coefficient $$\mathcal {F}_{n}(\delta X)$$ F n ( δ X ) of a Skorokhod-type stochastic differential $$\delta X_t=a(t,\omega )\,\delta B_t+b(t,\omega )\,\hbox {d}t$$ δ X t = a ( t , ω ) δ B t + b ( t , ω ) d t , where $$\mathcal {F}_{n}(\delta X)$$ F n ( δ X ) is defined by $$\mathcal {F}_{n}(\delta X):= \int _0^1 \overline{e_n(t)} a(t,\omega ) \,\delta B_t +\int _0^1 \overline{e_n(t)} b(t,\omega ) \, \hbox {d}t$$ F n ( δ X ) : = ∫ 0 1 e n ( t ) ¯ a ( t , ω ) δ B t + ∫ 0 1 e n ( t ) ¯ b ( t , ω ) 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.

Keywords: Stochastic Fourier coefficient; Skorokhod integral; Ogawa integral; Multiple Wiener–Itô integral; Brownian motion; Noncausal Wiener functional; Primary 60H05; Secondary 60H07 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0841-1

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