Posted on Categories:数学代写, 金融代写, 金融数学, 随机微积分

# 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|MAST90059 Natural FV Processes Are Predictable

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Natural FV Processes Are Predictable

The main result of this section is to show that a process $A \in \mathbb{V}$ is natural if and only if it is predictable. To achieve this, we need to consider the right continuous filtration $\left(\mathcal{F}_{.}^{+}\right)$along with the given filtration.

Recall that we had observed in Corollary $4.5$ that a $\left(\mathcal{F}^{+}\right)$predictable process $f$ such that $f_0$ is $\mathcal{F}0$ measurable is $\left(\mathcal{F}{.}\right)$predictable.

In his work on decomposition of submartingales, P. A. Meyer had introduced a notion of natural increasing process. It was an ad hoc definition, given with the aim of showing uniqueness in the Doob-Meyer decomposition.

Definition 8.31 Let $A \in \mathbb{V}0$; i.e. $A$ is an adapted process with finite variation paths and $A_0=0$. Suppose $|A|$ is locally integrable where $|A|_t=\operatorname{Var}{[0, t]}(A) . A$ is said to be natural if for all bounded r.c.l.I. martingales $M$
$[M, A]$ is a local martingale.
Let $\mathbb{W}=\left{V \in \mathbb{V}0:|V|\right.$ is locally integrable where $\left.|V|_t=\operatorname{VaR}{[0, t]}(V)\right}$.
Remark 8.32 Let $A \in \mathbb{V}_0$ be such that $[A, A]$ is locally integrable. Since $\Delta[A, A]$ $=(\Delta A)^2$, it follows that $(\triangle A)$ is locally integrable and as a consequence $A$ is locally integrable and thus $A \in \mathbb{W}$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Decomposition of Semimartingales Revisited

In view of this identification of natural $\mathrm{FV}$ processes as predictable, we can recast Theorem $5.50$ as follows.
Theorem $8.38$ Let $X$ be a stochastic integrator such that
(i) $X_t=X_{t \wedge T}$ for all $t$.
(ii) $\mathrm{E}\left[\sup {s \leq T}\left|X_s\right|\right]<\infty$. (iii) $\mathrm{E}\left[[X, X]_T\right]<\infty$. Then $\mathrm{X}$ admits a decomposition $$X=M+A, M \in \mathbb{M}^2, A \in \mathbb{V}, A_0=0 \text { and } A \text { is predictable } .$$ Further, the decomposition (8.4.1) is unique. Proof Let $X=M+A$ be the decomposition in Theorem 5.50. As seen in Corollary 5.50, the process $A$ satisfies $A_t=A{t \wedge T}$. Since $\mathrm{E}\left[[M, A]_T\right]=0$, we have
$$\mathrm{E}\left[[X, X]_T\right]=\mathrm{E}\left[[M, M]_T\right]+\mathrm{E}\left[[A, A]_T\right]$$
and hence $\mathrm{E}\left[[A, A]_T\right]<\infty$ and so $A \in \mathbb{W}$. Thus $A$ satisfies conditions of Theorem $8.37$ and hence $A$ is predictable. For uniqueness, if $X=N+B$ is another decomposition with $N \in \mathbb{M}^2$ and $B \in \mathbb{V}$ and $B$ being predictable, then $M-N=B-A$ is a predictable process with finite variation paths which is also a martingale and hence by Theorem $8.29, M=N$ and $B=A$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Doob-Meyer Decomposition

$$0 \leq C_t \leq A_t \quad \forall t$$

$$U=D+B$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales

$$M_t=\xi 1_{[\tau, \infty)}(t)$$

$$A_t=\mathrm{E}\left[\xi^2 \mid \mathcal{F} \tau-\right] 1[\tau, \infty)(t) .$$

$$M_t^2-A_t=\left(\xi^2-\mathrm{E}\left[\xi^2 \mid \mathcal{F}_\tau-\right]\right) 1[\tau, \infty)(t)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。