Speaker : Prof. Shigeyoshi OGAWA (Ritsumeikan University)
Title : Inversion in wide sense for natural SFT
Time : 2017-10-19 (Thu) 16:00 -
Place : Seminar Room 722, Institute of Mathematics (NTU Campus)
Abstract: Let $f(t, \omega)$ be a square integrable (in t $\in \left [ 0, 1 \right ]$) random function and $W_{t}(\omega)$ Brownian motion both defined on the same probability space. Given an orthonormal basis $\left \{ \varphi_{n} (t)\right \}$ in $L^{2}(0, 1)$, we introduce the random quantity $\hat{f}_n(\omega)$ by the following stochastic integral with respect to $W_{t}(\omega)$;
$\hat{f}_n(\omega) := \int_{0}^{1} f(t, \omega) \overline{\varphi_{n} (t)}dW_{t}, n \in N or Z.$

We call this ths SFC (stochastic Fourier coefficient) of $f(t, \omega)$ with respect to the basis $\left \{ \varphi_{n}\right \}$.
If the SFC $\hat{f}_n$ exists for every n and ${sup}_n$ $E\left [\left | \hat{f}_n\right |^{2}\right ] < \infty$ , then for such a random function $f(t, \omega)$ and an arbitrary $\displaystyle \ell^{2}$-sequence {$\epsilon_{n}$} such that $\epsilon_{n} \neq 0 \left (^{\forall }n \right )$ we see that the following random series
$\tau_{\epsilon,\varphi}$$f(t, \omega)$$ :=\underset{n}{\sum}$$\epsilon\hat{f}_n(\omega)\varphi_{n} (t).$

converges in $L^{2}([0, 1] × \Omega, dt × dP)$. We call this linear transform the SFT (stochastic Fourier transform) of $f(t, \omega)$ with respect to the pair ({$\epsilon_n$}, {$\varphi_{n}$}).We remark that the meaning of the SFT is not fixed until the stochastic integral used in the definition of SFC is specified and that we have many candidates for the integral. These notions of SFC and SFT were first introduced in 1986 by the author in the study of stochastic integral equation of Fredholm type ([2], [3]) and later, in 2013 found to have a deep connection with a numerical scheme for volatility estimation proposed by P.Malliavin $etal$ ([4],[5],[7]).
The aim of the talk is to present some very recent results about the fundamental question of inversibility of SFT, or almost equivalently of SFC. More precisely we are to show some direct inversion formulas of a special type SFT called $natural$ $SFT$ ([5]). The discussion will be developed in the framework of noncausal stochastic calculus ([1],[6]).

References
[1] Ogawa,S.: Sur le produit direct du bruit blanc par $l\hat{u}i-m\hat{e}me$., Comptes Rendus Acad Sci, Paris t.288, $S\acute{e}rie$ A, (1979) pp.359-362
[2] Ogawa,S. : On the stochastic integral equations of Fredholm type, in “Waves and Patterns” (monograph), Kinokuniya and North-Holland, (1986), pp.597-605
[3] Ogawa,S. : On a stochastic integral equation for the random fields, $S\acute{e}minaire$ de Proba., vol.25, Springer (1991), pp.324–339
[4] Ogawa,S. : Stochastic Fourier transformation, Stochastics 85-2 (2013), pp.286–294
[5] Ogawa,S.: On a direct inversion formula for the SFT, Sankhya 22 (2015)
[6] Ogawa,S. : “Noncausal Stochastic Calculus”, (monograph) Springer Verlag (2017) August
[7] Malliavin,P. and Thalmeyer,A. : ”Stochastic calculus of variations in mathematical finance”, Springer-Verlag (2009)