臺灣博碩士論文加值系統

在這篇論文中, 我們建構 martingale 在白噪音空間上. 如果
$a,\,b\in\mathbb{C}$ 且 $a^2+b^2=0$, 然後我們可得 對
$\calF_{t}=\{B(s)~; ~0\leq s\leq t\}$ 而言,
$g_{ab}(\varphi\circ\theta_{t})(x)$ 是 martingale . 其中 $g_{ab}$ 是
Fourier-Gauss transforms, 其定義為
$g_{ab}\varphi(x)=\int_{S^{*}}\varphi(ay+bx)\mu(dy)$. 令
$\theta_{t}$ 是一個算子, 其定義為對任意 $x\in L^{2}(\r)$
$$\theta_{t}x(s)=1_{(-\infty,t]}(s)x(s).$$
~~令 $\{ ~e_{j}:1\leq j\leq n \}$ 是一組正交規範集存在於
$L^{2}(\r^{1})$ , 而且 ${\calF}_{n} =$$\sigma$ \{$(x,e_{j})$ :
$1\leq j\leq n$\} . 如果 $P_{n}$ 為 $L^{2}(\r^{1})$ 映成至由
$\{~e_{j}: 1\leq j \leq n \}$ 所生成的空間之正交投影.
我們可得對$\calF_{n}$而言, $g_{ab}(\varphi\circ P_{n})$ 是
martingale , 若且唯若 $a^{2}+b^{2}=0$.
此外, 我們可以藉由 martingale 在 Test
函數上的觀念推廣到廣義函數上. 最後在應用方面我們給一些例子來驗證.

The main goal of this paper is to construct martingales on white
noise space. It is shown that, if $a,\,b\in\mathbb{C}$ such that
$a^2+b^2=0$, then $g_{ab}(\varphi\circ\theta_{t})(x)$ is a
martingale with respect to the filtration $\calF_{t}$, where
$\calF_{t}$ is the Borel field generated by $\{B(s)~; ~0\leq s\leq
t\}$,
$g_{ab}$ the Fourier-Gauss
transforms defined by
$g_{ab}\varphi(x)=\int_{S^{*}}\varphi(ay+bx)\mu(dy)$, for any entire
function $\varphi$ of exponential growth of order two, and
$\theta_{t}$ is an operator defined by
$$\theta_{t}x(s)=1_{(-\infty,t]}(s)x(s).$$
for all $x\in L^{2}(\r)$. Let $\{ ~e_{j}:1\leq j\leq n \}$ be complete orthonormal set
of
$L^{2}(\r^{1})$ , and ${\calF}_{n} =$$\sigma$ \{$(x,e_{j})$ : $1\leq
j\leq n$\} ($(x,e_{j})=\int_{\r}e_{j}(t)dB(t)$) and $P_{n}$ the
orthogonal projection of $L^{2}(\r^{1})$ onto the space spanned by
$\{~e_{j}: 1\leq j \leq n \}$. It is shown that $g_{ab}(\varphi\circ
P_{n})$ is martingale with respect to $\calF_{n}$, if and only if
$a^{2}+b^{2}=0$. Further, we extend the concept of martingale to
generalized white noise functions. As applications, we give some
examples for verifying our theorem .

1.Introduction......................4
2.Preliminaries.....................5
3.Main Results......................9
4.Generalized Martingale...........12
5.Example...........................13

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