Keywords: stochastic integral, spectral representation
定义在正交增量随机过程上的随机积分算子是一个测度为F的Hilbert空间的同构(或者说具有Borel集的拓扑空间)， 且这个测度F和正交增量随机过程是一一对应的。我们可以证明这个同构well-defined。这里都是基于某个特定的随机变量的内积定义来的。如果我们换了随机变量的内积，这里都要变化（例如，我们要研究一个更大的Hilbert空间，例如$L^1$, 即强大数率的tech条件）。
1. Definitions
Again, let’s consider a $L^2[-\pi, \pi]$ Hilbert space, whose inner product of random variable $X$ and $Y$ are defined as $\mathbb{E}(X\bar{Y})$, and $\int_{\Omega} X(\omega) \bar{Y}(\omega)P(d\omega)$
1.1 Orthogonal-increment process
An orthogonal-increment process on $[-\pi, \pi]$ is a complex-valued stochastic process $\{Z(\lambda), -\pi \le \lambda \le \pi\}$ such that $(Z(\lambda), Z(\lambda)) < \infty$, $(Z(\lambda), 1) = 0$, $(Z(\lambda_4) - Z(\lambda_3), Z(\lambda_2) - Z(\lambda_1)) = 0$ if $(\lambda_1, \lambda_2] \cap (\lambda_3, \lambda_4] = \phi$
1.2 Right-continuous process
The process $\{Z(\lambda, -\pi \le \lambda \le \pi)\}$ will be called right-continuous if for all $\lambda \in [-\pi, \pi)$
$|Z(\lambda + \delta) - Z(\lambda)|^2 = \mathbb{E}(|Z(\lambda + \delta) - Z(\lambda)|^2) \xrightarrow{} 0$ as $\lambda \xrightarrow{} 0 $
Proposition 1.1: If $\{Z(\lambda), -\pi < \lambda < \pi\}$ is an orthogonal-increment process, then there is a unique distribution function $F$ (i.e. a unique non-decreasing, right-continuous function) such that
$F(\lambda) = 0, \lambda \le -\pi$
$F(\lambda) = F(\pi), \lambda \ge \pi$
$F(\mu) - F(\lambda) = |Z(\mu) - Z(\lambda)|^2$
Proof :(构造法证明)
$F(\mu) = |Z(\mu) - Z(-\pi)|^2$, $-\pi \le \mu \le \pi$
According to the orthogonality of $Z(\mu) - Z(\lambda)$ and $Z(\lambda) - Z(-\pi)$, $-\pi \le \lambda \le \mu \le \pi$
$F(\mu) = |Z(\mu) - Z(\lambda) + Z(\lambda) - Z(-\pi)|^2 = |Z(\mu) - Z(\lambda)|^2 + |Z(\lambda) - Z(-\pi)||^2 \ge |Z(\lambda) - Z(-\pi)|^2 = F(\lambda)$
$F(\mu + \delta) - F(\mu) = |Z(\mu + \delta) - Z(\mu)|^2 \xrightarrow{} 0$ as $\delta \xrightarrow{} 0$
Remark： In many field the properties of proposition 1.1 will be written as $\mathbb{E}(dZ(\lambda)d\bar{Z}(\mu)) = \delta_{\lambda, \mu} dF(\lambda)$
2. Stochastic Integration wrt an Orthogonal Increment Process
Suppose $\{Z(\lambda), -\pi \le \lambda \le \pi\}$ is an orthogonal-increment process defined on the probability space $(\Omega, \{\mathcal{F}\}_{\lambda}, P)$. Note that the inner-product defined on this Hilbert space is:
$(Z_{\lambda}, Z_{\lambda}) = \int_{\Omega}Z_\lambda(\omega) P(d\omega)$
Let $f$ be a Borel measurable function on complex-valued space Hilbert space $([-\pi, \pi], \mathcal{B}, F)$. Let $I(f)$ denote $\int_{(-\pi, \pi])}f(\lambda) dZ(\lambda)$ , which is a random variable of the $L^2(\Omega, \mathcal{F}, P)$ (of course there exist some technical condition like $\int_{(-\pi, \pi])} f(\lambda)^2 dF(\lambda)$ to make sure this random variable is square integrable. We can easily verify this condition based on the isomorphism later). We want to solve the unknown measure $F(\lambda)$, such that $(I(f), I(g)) = \int_{(-\pi, \pi])} f(\lambda)\bar{g}(\lambda) dF(\lambda)$
Note that any measurable function can be written as the summation of simple function. Therefore, let $f(\lambda) = \sum_{i = 0}^n f_i I_{(\lambda_i, \lambda_{i + 1}]}(\lambda)$, where $-\pi = \lambda_0 < \lambda_1 < \dots < \lambda_{n + 1} = \pi$. Therefore, we can rewrite the stochastic integral $I(f) = \int_{(-\pi, \pi]}f(v)dZ(v)$ as $I(f) = \sum_{i = 0}^n f_i[Z({\lambda_{i + 1}}) - Z(\lambda_{i})]$.
The measure F can be solved based on the definition of isomorphism can be shown as:
$(I(f), I (g)) = (\sum_{i = 0}^{n}f_i[Z(\lambda_{i + 1}) - Z(\lambda_{i})], \sum_{i = 0}^{n}g_i[Z(\lambda_{i + 1}) - Z(\lambda_{i})]) = \sum_{i = 0}^{n} f_i\bar{g}_i(F(\lambda_{i + 1} - F(\lambda_i)))$
The measure $F$ is nothing but the associated distribution function we defined in the proposition 1.1.
Remark: $F$ is unique due to the proposition 1.1. The continuous version can be written as $\int_{-\pi, \pi} f(v)\bar{g}(v)dF(v) = (f, g)_{L^2(F)}$
We can further prove the technical condition of $f$ that $I(f)$ belongs to $L^2(\Omega, \mathcal{F}, P)$, and verify that $F$ is the measure of a Hilbert space. (我们必须要证明这样的一个Hilbert空间是存在的，或者说是well-defined。 这种well-defined必须通过概率空间的well-defined证明。这里的过程虽然在逻辑上重要，但是在应用上可以忽略）
2.1 Definition of stochastic integral
If $\{Z(\lambda)\}$ is an orthogonal-increment process on $[-\pi, \pi]$ with associated distribution function $F$ and if $f \in L^2(F)$ (F is constructed in Proposition 1.1), then the stochastic integral $\int_{(-\pi, \pi])} f(\lambda) d Z(\lambda)$ is defined as the random variable $I(f)$ constructed above, i.e.
$\int_{(-\pi, \pi]} f(v)dZ(v) = I(f)$
2.2 Properties of the Stochastic Integral
(a) The stochastic integral operator $I$ is a linear operator, i.e, $I(af + bg) = aI(f) + bI(g)$
(b) Isometry: $\mathbb{E}(I(f)\bar{I(g)}) = \int_{(-\pi, \pi])}f(v)\bar{g}(v)dv$
(c) $\mathbb{E}(I(f)) = 0$ (正交增量过程的期望是0)
Note that if $Z(\lambda)$ is any orthogonal increment process on $[-\pi, \pi]$ with associated distribution function $F$, then
$X_t = I(e^{it \cdot}) = \int_{(-\pi, \pi]} e^{itv} dZ(v)$ $t \in \mathbb{Z}$
is a stationary process with mean zero and autocovariance function:
$\mathbb{E}(X_{t + h}\bar{X}_t) = \int_{(-\pi. \pi]} e^{iv(t+h)}e^{-ivt}dF(v) = \int_{(-\pi. \pi]} e^{ivh}dF(v)$
Note that the converse also holds: if $\{X_t\}$ is any stationary process, then $\{X_t\}$ has the representation for an appropriately chosen orthogonal process whose associated distribution function is the same as the spectral distribution function of $\{X_t\}$
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