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# 数学代写|随机分析作业代写stochastic analysis代考|Several Notions on Stochastic Processes

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## 数学代写|随机分析作业代写stochastic analysis代考|complete probability space

Let $(\Omega, \mathcal{F}, P)$ be a complete probability space, that is, any subset $B$ of $\Omega$ for which there is an $A \in \mathcal{F}$ such that $B \subset A$ and $P(A)=0$ belongs to $\mathcal{F}$.

Let $\mathcal{N}{0}$ be the set of $A \in \mathcal{F}$ such that $P(A)=0$ or 1 . Then $\mathcal{N}{0}$ is a sub- $\sigma$-algebra. In this section, we consider the case that $\mathbf{T}=[0, \infty)$.

Definition 3.1.1 We say that a filtration $\left{\mathcal{F}{t}\right}{t \in[0, \infty)}$ satisfies the standard condition, if the following two conditions are satisfied.
(1) $\mathcal{N}{0} \subset \mathcal{F}{0}$.
(2) (Right continuity) $\bigcap_{s>t} \mathcal{F}{s}=\mathcal{F}{t}$ for any $t \geqq 0$.
We say that $\left(\Omega, \mathcal{F}, P,\left{\mathcal{F}{t}\right}{t \in[0, \infty)}\right)$ is a standard filtered probability space, if $(\Omega, \mathcal{F}, P)$ is a complete probability space and a filtration $\left{\mathcal{F}{t}\right}{t \in[0, \infty)}$ satisfies the standard condition.

We assume that $\left(\Omega, \mathcal{F}, P,\left{\mathcal{F}{t}\right}{t \in[0, \infty)}\right)$ is a standard filtered probability space from now on.

A stochastic process $X=\left{X_{t}\right}_{t \in[0, \infty)}$ is a family of random variables. We regard $X$ as a function defined in $[0, \infty) \times \Omega$. We introduce several notions in the following in order to analyze stochastic processes.

## 数学代写|随机分析作业代写stochastic analysis代考|stochastic process

Proposition 3.1.2 Let $X:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ be an $\left(\left{\mathcal{F}{t}\right}{\left.t \in[0, \infty)^{-}\right)}\right.$adapted process, and that
$$E\left[\left|X_{t}\right|\right]<\infty, \quad t \geqq 0$$ Let $\left{t_{n}\right}_{n=0}^{\infty}$ is a sequence of increasing non-negative numbers such that $t_{0}=0, t_{n-1}<$ $t_{n}, n \geqq 1$, and $t_{n} \rightarrow \infty, n \rightarrow \infty$. Suppose that for any $n \geqq 1$ $$E\left[X_{t} \mid \mathcal{F}{s}\right]=X{s}, \quad s, t \in\left[t_{n-1}, t_{n}\right], ss \geqq 0 such that t \in\left[t_{n-1}, t_{n}\right], s \in\left[t_{m-1}, t_{m}\right], n \geqq m \geqq 1. We prove this claim by induction in n-m. From the assumption we see that the claim is valid in the case that n-m=0. Assume that the claim is valid in the case that n-m=k \geqq 0. Suppose that n-m=k+1. Since t_{m} \in\left[t_{m}, t_{m+1}\right] and n-(m+1)=k, we see that$$
E\left[X_{t} \mid \mathcal{F}{s}\right]=E\left[E\left[X{t} \mid \mathcal{F}{t{m}}\right] \mid \mathcal{F}{s}\right]=E\left[X{t_{m}} \mid \mathcal{F}{s}\right]=X{s}
$$So the claim is shown. In this book, we use the following notation. \Delta_{n}, n \geqq 1, and \Delta are sets defined by$$
\Delta_{n}=\left{\frac{k}{2^{n}} ; k \in \mathbf{Z}{\geq 0}\right}, \quad n \geqq 1, \quad \text { and } \Delta=\bigcup{n=1}^{\infty} \Delta_{n}
$$## 随机分析代写 ## 数学代写|随机分析作业代写STOCHASTIC ANALYSIS代考|COMPLETE PROBABILITY SPACE 让(Ω,F,磷)是一个完整的概率空间，即任何子集乙的Ω有一个一种∈F这样乙⊂一种和磷(一种)=0属于F. Let \mathcal{N}{0} be the set of A \in \mathcal{F} such that P(A)=0 or 1 . Then \mathcal{N}{0} is a sub- \sigma-algebra. In this section, we consider the case that \mathbf{T}=[0, \infty). 定义 3.1.1 我们说一个过滤 \left{\mathcal{F}{t}\right}{t \in[0, \infty)} satisfies the standard condition, if the following two conditions are satisfied. (1) \mathcal{N}{0} \subset \mathcal{F}{0}. (2) (Right continuity) \bigcap_{s>t} \mathcal{F}{s}=\mathcal{F}{t} for any t \geqq 0. We say that \left(\Omega, \mathcal{F}, P,\left{\mathcal{F}{t}\right}{t \in[0, \infty)}\right) is a standard filtered probability space, if (\Omega, \mathcal{F}, P) is a complete probability space and a filtration \left{\mathcal{F}{t}\right}{t \in[0, \infty)} .满足标准条件。 我们假设 \left(\Omega, \mathcal{F}, P,\left{\mathcal{F} {t}\right} {t \in[0, \infty)}\right) 是一个标准从现在开始过滤概率空间。 随机过程X=\left{X_{t}\right}_{t \in[0, \infty)}X=\left{X_{t}\right}_{t \in[0, \infty)}是一个随机变量族。我们认为X作为定义的函数[0,∞)×Ω. 为了分析随机过程，我们在下面介绍几个概念。 ## 数学代写|随机分析作业代写STOCHASTIC ANALYSIS代考|STOCHASTIC PROCESS 命题 3.1.2 Let X:[0, \infty) \times \Omega \rightarrow \mathbf{R} be an \left(\left{\mathcal{F}{t}\right}{\left.t \in[0, \infty)^{-}\right)}\right.adapted process, and that$$
E\left[\left|X_{t}\right|\right]<\infty, \quad t \geqq 0 $$Let \left{t_{n}\right}_{n=0}^{\infty} is a sequence of increasing non-negative numbers such that t_{0}=0, t_{n-1}< t_{n}, n \geqq 1, and t_{n} \rightarrow \infty, n \rightarrow \infty. Suppose that for any n \geqq 1$$ E\left[X_{t} \mid \mathcal{F}{s}\right]=X{s}, \quad s, t \in\left[t_{n-1}, t_{n}\right], ss \geqq 0$such that$t \in\left[t_{n-1}, t_{n}\right], s \in\left[t_{m-1}, t_{m}\right]$,$n \geqq m \geqq 1$. 我们通过归纳证明这一主张n−米. 从假设中，我们看到该主张在以下情况下是有效的n−米=0. 假设索赔在以下情况下是有效的n−米=到≧0. 假设n−米=到+1. 自从吨米∈[吨米,吨米+1]和n−(米+1)=到, 我们看到 $$E\left[X_{t} \mid \mathcal{F}{s}\right]=E\left[E\left[X{t} \mid \mathcal{F}{t{m}}\right] \mid \mathcal{F}{s}\right]=E\left[X{t_{m}} \mid \mathcal{F}{s}\right]=X{s}$$ So the claim is shown. In this book, we use the following notation.$\Delta_{n}, n \geqq 1$, and$\Delta\$ are sets defined by
$$\Delta_{n}=\left{\frac{k}{2^{n}} ; k \in \mathbf{Z}{\geq 0}\right}, \quad n \geqq 1, \quad \text { and } \Delta=\bigcup{n=1}^{\infty} \Delta_{n}$$