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# 数学代写|随机过程Stochastic Porcesses代考|AMATH562 Wold Decomposition Theorem (1938)

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## 数学代写|随机过程代写Stochastic Porcesses代考|Wold Decomposition Theorem (1938)

Any stationary process $X(t)$ can be expressed in the form $X(t)=U(t)+V(t)$,
(a) where $U(t)$ and $V(t)$ are uncorrelated proces.
(b) $U(t)$ is regular (or indeterministic) with a one-sided linear representation
$$U(t)=\sum_{i=0}^{\infty} c_i \varepsilon_{t-i} \text { with } c_0=1, \sum_{i=0}^{\infty} c_i^2<\infty$$
and $\varepsilon_t$ is white noise W.N. $\left(0, \sigma^2\right)$ and uncorrelated with $V(t)$, i.e. $E\left(\varepsilon_t V(s)\right)=0$ for all $t, s$.
The sequence $\left{c_i\right}$ and the process $\left{\varepsilon_t\right}$ are uniquely determined.
(c) $V(t)$ is singular (or determinstic) i.e. can be predicted from its own past with zero prediction variance (See Note 1, page 286).

Proof Let $\hat{X}(t)$ denote the projection of $X(t)$ on $\chi_{t-1}$ (i.e. the linear subspace of $\chi$ spanned by the r.vs. $X(s), s \leq t-1$ i.e. limits of Cauchy sequences of linear combinations of $X_{t-1}, X_{t-2}, \ldots$ and the all such linear combinations. In fact $\hat{X}(t)$ is the linear least square predictor of $X(t)$ given $X(t-1), X(t-2), \ldots)$. Define the process $\left{\varepsilon_t\right}$ by $\varepsilon_t=X(t)-\hat{X}(t)$. Then $\varepsilon_t \perp \chi_{t-1}$ i.e. $\varepsilon_t$ is orthogonal to every element of $\chi_{i-1}$. For any pair, $\left{\varepsilon_s, \varepsilon_t\right}$ with, say $s<t$, We have $\varepsilon_t \perp \chi_s \subset \chi_{t-1}$, and $\varepsilon_s \in \chi_s$ (since both $X(t)$ and $\hat{X}(s)$ are in $\chi_s$ ).

## 数学代写|随机过程代写Stochastic Porcesses代考|Introduction

In recent years, it has become apparent that physical systems, classically modelled by deterministic differential equations, can be more satisfactorily modelled by certain stochastic counterparts if random effects in the physical phenomena as well as measuring devices are to be taken into account. In this context, an ordinary differential equation $\frac{d x}{d t}=f(t, x)$ would be replaced by a random differential
$$\frac{d X(t)}{d t}=F(t, X(t), Y(t))$$
where $Y(t)$ represent some stochastic input process explicitly. It is often seen that it is not possible to interpret (A.III.1) as an ordinary differential equation along each sample path. For example
$$\frac{d X(t)}{d t}=f(t, X(t))+g((t, X(t)) N(t)$$
with $N(t)$ being a Gaussion white noise process. Equation (A.III.2) has been popular in the engineering literature especially, since Gaussion white noise approximates the effect of the superposition of a large number of small random disturbances, a situation encountered in engineering systems. However, the irregularity of the sample paths of $N$ makes (A.III.2) intractable mathematically. Just as a solution of (A.III.1) satisfies the deterministically integral equation $X(t)=X\left(t_0\right)+\int_{t_0}^t f(s, X(s)) d s$ a solution of (A.III.2) should be a solution of the random equation
$$X(t)=X\left(t_0\right)+\int_{t_0}^t f(s, X(s)) d s+\int_{t_0}^t g(s, X(s)) N(s) d s$$
but unfortunately the last integral in (A.III.3) cannot be defined in any meaningful way. To deal with this difficulty, the integral in question is replaced by an integral of the form
$$\int_{t_0}^t g(s, X(s)) d W(s) .$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Wold Decomposition Theorem (1938)

(a) 其中 $U(t)$ 和 $V(t)$ 是不相关的过程。
（二） $U(t)$ 是规则的（或不确定的），具有兰边线性表示
$$U(t)=\sum_{i=0}^{\infty} c_i \varepsilon_{t-i} \text { with } c_0=1, \sum_{i=0}^{\infty} c_i^2<\infty$$

(C) $V(t)$ 是奇异的（或确定性的），即可以用零预则方差从它自己的过去预测（见注释 1 ，第 286 页）。

left 缺少或无法识别的分隔符 $\quad$ 经过 $\varepsilon_t=X(t)-\hat{X}(t)$. 然后 $\varepsilon_t \perp \chi_{t-1} \mathrm{IE} \varepsilon_t$ 正交于的每个元絠 $\chi_{i-1}$. 对于任 何一对，〈left 缺少或无法识别的分隔符 $\quad 与$ 与说 $s<t$ ，我们有 $\varepsilon_t \perp \chi_s \subset \chi_{t-1}$ ，和 $\varepsilon_s \in \chi_s$ (因为两者 $X(t)$ 和 $\hat{X}(s)$ 在 $\left.\chi_s\right)$.

## 数学代写|随机过程代写Stochastic Porcesses代考|Introduction

$$\frac{d X(t)}{d t}=F(t, X(t), Y(t))$$

$$\frac{d X(t)}{d t}=f(t, X(t))+g((t, X(t)) N(t)$$

$$X(t)=X\left(t_0\right)+\int_{t_0}^t f(s, X(s)) d s+\int_{t_0}^t g(s, X(s)) N(s) d s$$
$$\int_{t_0}^t g(s, X(s)) d W(s) .$$

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