Direct Inversion Formulas for the Natural SFT

Shigeyoshi Ogawa ()
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Shigeyoshi Ogawa: Ritsumeikan University

Sankhya A: The Indian Journal of Statistics, 2018, vol. 80, issue 2, No 4, 267-279

Abstract: Abstract 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; T 𝜖 , φ f ( t , ω ) : = ∑ n 𝜖 n f ̂ n ( ω ) φ n ( t ) ${\mathcal T}_{\epsilon , \varphi }f(t,\o ):= {\sum }_{n} \epsilon _{n} \hat {f}_{n}(\o )\varphi _{n}(t)$ where {𝜖n} is an ℓ2-sequence such that 𝜖n ≠ 0, ∀n and f ̂ n $\hat {f}_{n}$ is the SFC (short for “stochastic Fourier coefficient”) defined by f ̂ n ( ω ) = ∫ 0 1 f ( t , ω ) φ n ( t ) ¯ d W t $\hat {f}_{n}(\o )={{\int }_{0}^{1}} f(t,\o )\overline {\varphi _{n}(t)}dW_{t}$ , 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: Brownian motion; Stochastic integrals; Fourier series; Primary: 60H05; 60H07; Secondary: 42A61 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s13171-018-0128-8

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