臺灣博碩士論文加值系統

本論文中，證明在給定布朗運動 B(t) 的情況下，白雜訊泛函 $\varphi$ 之條件期望值可以表示為$$E[\varphi|\mathcal{B}_{t}]=\int_{S^*}\varphi[\Theta_tx+(1-\Theta_t)y] \mu(dy),$$其中 $\Theta_t$ 是Heaviside函數
$$\Theta_t(s)\equiv \left\{\begin{array}{ll}\mbox{I}& ,s\leq t ,\cr 0 &,s>t.
\end{array}\right.$$
且 $\Theta_t$ 是Heaviside算子其定義為 $\Theta_{t}x(s)=\Theta_{t}(s)x(s)$.
我們可以將布朗運動表示成以下形式
$$B_{t}(x)=\left\{\begin{array}{rr}\langle x,1_{[0,t]}\rangle & , t\geq 0, x\in S^*(\mathbb{R}^1) ;\cr-\langle x,1_{[t,0]}\rangle & , t< 0, x\in S^*(\mathbb{R}^1).\end{array}\right.$$如果 $\{e_{j}:1\leq j\leq n \}$ 是一個正交規範集存在於 $L^2(\mathbb{R}^1)$ 而且 $\mathcal{B}_{n}=\sigma \{\langle
x,e_{j}\rangle:1\leq j \leq n\}$。如果 $P_{n}$ 為$L^2(\mathbb{R}^1)$ 映成至由 $\{e_{j}:1\leq j\leq n \}$所生成的空間之正交投影。我們證明條件期望值可以用積分形式表示為$$E[\varphi|\mathcal{B}_{n}]=\int_{S^*}\varphi[P_{n}x+(1-P_{n})y]\mu(dy)$$利用以上的積分表示式，我們可以用來研究條件期望值的正則性質而且提供簡易的計算方式。此外，我們可以延伸條件期望值的概念到廣義的白雜訊泛函。最後在應用方面，我們給一些例子來說明。

In this paper it is show that the conditional
expectation of a white noise functional $\varphi$ given the the
Brownian motion $B(t)$ is represented by
$$E[\varphi|\mathcal{B}_{t}]=\int_{S^*}
\varphi[\Theta_tx+(1-\Theta_t)y] \mu(dy)\ ,$$
where $\Theta_t$ is the Heaviside function
$$\Theta_t(s)\equiv \left\{
\begin{array}{ll}
\mbox{I}& ,s\leq t ,\cr 0 &,s>t.
\end{array}\right.$$
and $\Theta_t$ is the Heaviside operator defined by
$\Theta_{t}x(s)=\Theta_{t}(s)x(s)$. Note that the Brownian motion $B(t)$ can be represented by
$$B_{t}(x)=\left\{
\begin{array}{rr}
\langle x,1_{[0,t]}\rangle & , t\geq 0, x\in S^*(\mathbb{R}^1) ;\cr
-\langle x,1_{[t,0]}\rangle & , t< 0, x\in S^*(\mathbb{R}^1).
\end{array}\right.$$
If $\{e_{j}:1\leq j\leq n \}$ be an orthonormal set in
$L^2(\mathbb{R}^1)$ and $\mathcal{B}_{n}=\sigma \{\langle
x,e_{j}\rangle:1\leq j \leq n\}$ and if $P_{n}$ denotes the
orthogonal projection of $L^2(\mathbb{R}^1)$ onto the space spanned
by $\{e_{j}:1\leq j\leq n \}$, then it is shown that conditional
expectation enjoy the integral representation
$$E[\varphi|\mathcal{B}_{n}]=\int_{S^*}\varphi[P_{n}x+(1-P_{n})y]\mu(dy)$$
Using the above integral representation we are able to investigate
the regularity properties of the conditional expectation and compute
the conditional expectation easily. Moreover, we can extend the
concept of conditional expectation to generalized white noise
functionals. As applications, we give some examples.

1 Introduction 1
2 Preliminaries 2
3 The Schwartz Space as a nuclear space 3
4 Test and Generalized White Noise Functionals 4
5 Conditional Expectations of GWNF 6
6 Example 10
References 14

[1] L. Gross, Abstract Wiener spaces, in ”Proceedings 5th Berkeley Symp. Math.
Stat. Prob. ,1965” Vol.2, pp.31-42.
[2] L. Gross, Potential Theory on Hilbert Space, J. Funct. Anal. 1(1967), 123-181.
[3] K. Ito, Multiple Wiener integral, J. Math. Soc. Japan 3(1951), 157-169.
[4] T. Hida, Analysis of Brownian functionals, in ”Carleton Math. Lecture Notes
No. 13” Carleton Univ. Ottawa, 1975.
[5] T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise: An Infinite Dimensional
Calculus, Kluwer, Dordrecht/Boston/London, 1993.
[6] H.-H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J.
41(1974), 373-379.
[7] H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996.
[8] H.-H. Kuo, Gaussian Measure in Banach Spaces, in ”Lecture Notes in Math.
Vol. 463,” Springer-Verlag, Berlin/New York, 1975.
[9] H.-H. Kuo, Distribution theory on Banach Space, in ”Lecture Notes in Math.
Vol. 526,” pp.143-156, Springer-Verlag, Berlin/New York, 1976.
[10] H.-H. Kuo, On Fourier Transform of Generalized Brownain Functionals, J. Multivariate
Anal. 12(1982), 415-431.
[11] H.-H. Kuo, Donsker’s delta function as a generalized Brownian functional and
its application, in ”Lecture Notes in Control and Information Sciences Vol. 49,”
pp.167-178 Springer-Verlag, Berlin/New York, 1983.
[12] H.-H. Kuo, Brownian motion, diffusions and infinite dimensional calculus, Notes
delivered in the workshop on infinite dimensional calculus and its applications
at Math. Research Center, Taiwan Univ., July-August, 1986.
[13] H.-H. Kuo, and A. Russek, White Noise Approach to Stochastic Integration, J.
Multivariate Anal. 24(1988), 218-236.
[14] Y.-J. Lee, A characterization of generalized functions on infinite dimensional
spaces and Bargmann-Segal analytic functions, in ”Gaussian Random Field,”
the third Nagoya L´evy Seminar (K. Itˆo and T. Hida, Eds.), pp.272-284, World
Scientific, Singapore, 1991.
[15] Y.-J. Lee, A reformulation of white noise calculus, in ”White Noise Analysis”
(T. Hida et al.,Eds.), pp.274-293, World Scientific, Singapore, 1990.
[16] Y.-J. Lee, Applications of the Fourier-Wiener transform to differential equations
on infinite dimensional spaces I, Trans. Amer. Math. Soc. 262(1980), 259-283.
[17] Y.-J. Lee, Analytic version of test functionals, Fourier transform, and a characterization
of measures in white noise calculus, J. Funct. Anal. 100(1991), 359-380.
[18] Y.-J. Lee, Calculus of generalized white noise functionals, Proc. Preseminar of
International Conference on Gaussian Random Fields, 66-125, Department of
Mathematics, Nagoya Univ., Nagoya, Japan, 1990.
[19] Y.-J. Lee, Generalized function on infinite dimensional spaces and its application
to white noise calculus, J. Funct. Anal. 82(1989), 429-464.
[20] Y.-J. Lee, Integral representation of second quantization and Its application to
white noise analysis, J. Funct. Anal. 2(1995), 253-276.
[21] Y.-J. Lee, Integral transform of analytic functions on abstract Wiener space, J.
Funct. Anal. 47(1982), 153-164.
[22] Y.-J. Lee, On the convergence of Wiener-Itˆo decomposition, Bull. Inst. Math.
Academia Sinica, 17(1989), 305-312.
[23] Y.-J. Lee, Transformation and Wiener-Ito decomposition of white noise functionals,
Bull. Inst. Math. Academia Sinica (Taiwan), 21(1993), 279-291.
[24] Y.-J. Lee, H.-C. Huang, Conditional expectation of Generalized Wiener Functionals.
[25] M. de Faria, M. J. Oliverira, and L. Streit, A Generalized Clark-Ocone Formula,
preprint, 1999.
[26] I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 2, Academic Press,
New York/London, 1964.
[27] I.M. Gelfand and N.Ya. Vilenkin, Generalized Functions, Vol. 4, Academic Press,
New York/London, 1964.