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## 统计代写|鞅论代写Martingale Theory代考|The Frictionless Market Assumption

For subsequent usage, a market is defined to be collection $((\mathbb{B}, \mathbb{S}), \mathbb{F}, \mathbb{P})$ representing the stochastic processes for the traded assets, the market’s information set, and the statistical probability measure. The underlying state space and $\sigma$-algebra, $(\Omega, \mathscr{F})$, are always implicit in this collection and not included. A market is always assumed to be frictionless, with the exception of the admissibility condition (unless otherwise indicated), and competitive.

The assumption of frictionless markets implicitly appears in the definition of the set of admissible s.f.t.s. $\mathscr{A}(x)$ in that, except for the admissibility constraint, the accumulated value of the trading strategy has no additional adjustments for other frictions, e.g. transaction costs, taxes, indivisible shares, explicit short sale constraints, or margin requirements. As noted previously, the admissibility condition is, in fact, a trading constraint imposed on the aggregate value of all shorts in the s.f.t.s.

The assumption of competitive markets also implicitly appears in the definition of $\mathscr{A}(x)$ because the price processes $\left(\mathbb{B}_u, \mathbb{S}_u\right)$ do not depend on the trading strategy $\left(\alpha_0, \alpha\right)$. The trader is a price-taker because there is no quantity impact on the price processes from trading the shares $\left(\alpha_0, \alpha\right)$.

Although a misnomer, the convention in the literature is to still call a market frictionless if the only restriction imposed is the admissibility condition. Because admissibility is needed to exclude doubling strategies, its imposition is thought to be very mild. It is also standard in portfolio optimization problems, in the context of a frictionless market, to impose an analogous constraint that a trader’s wealth is always nonnegative (see Part II of this book).

For the remainder of the book, a market is always assumed to be frictionless in the sense just discussed. It is important to keep this misnomer in mind when using the phrase “frictionless markets” in the subsequent models. Doing so one can more easily understand why asset price bubbles often exist as an implication of the model. For example, in the asset price bubbles Chap. 3 this clarifies why asset price bubbles exist in a frictionless and competitive market where there are no arbitrage opportunities (to be defined). Second, in the portfolio optimization Chaps. 10-12, this clarifies how an optimal wealth and consumption path can exist in the presence of asset price bubbles in a frictionless and competitive market. And finally, in Chaps. 13-16 that study economic equilibrium, this also clarifies how asset price bubbles can exist in a frictionless and competitive market rational equilibrium.

## 统计代写|鞅论代写Martingale Theory代考|Change of Numeraire

Normalization by the money market account, which is a change of numeraire, simplifies the notation and is almost without loss of generality. The lost of generality is that the set of trading strategies $\mathscr{A}(x)$ after the change of numeraire may differ from the set of trading strategies before due to the modified integrability conditions needed to guarantee that the relevant integrals exist. This section presents the new notation and the evolutions for the mma and the risky assets under this change of numeraire.

Let $B_t=\frac{\mathbb{B}_t}{\mathbb{B}_t}=1$ for all $t \geq 0$, this represents the normalized value of the money market account (mma).

Let $S_t=\left(S_1(t), \ldots, S_n(t)\right)^{\prime} \geq 0$ represent the risky asset prices when normalized by the value of the mma, i.e. $S_i(t)=\frac{\mathbb{S}_i(t)}{\mathbb{B}_t}$. Then,
$$\begin{gathered} \frac{d B_t}{B_t}=0 \quad \text { and } \ \frac{d S_t}{S_t}=\frac{d \mathbb{S}_t}{\mathbb{S}_t}-r_t d t \end{gathered}$$
Proof Using the integration by parts formula Theorem 3 in Chap. 1, one obtains (dropping the t’s)
$$d\left(\frac{\mathbb{S}}{\mathbb{B}}\right)=\frac{1}{\mathbb{B}} d \mathbb{S}+\mathbb{S} d\left(\frac{1}{\mathbb{B}}\right)=\frac{d \mathbb{S}}{\mathbb{S}} \mathbb{B}-\frac{\mathbb{S}}{\mathbb{B}} \frac{d \mathbb{B}}{\mathbb{B}} .$$
The first equality uses $d\left[\mathbb{S}, \frac{1}{\mathbb{B}}\right]=0$, since $\mathbb{B}$ is continuous and of finite variation (use Lemmas 2 and 7 in Chap. 1).
Substitution yields
$d S=\frac{d S}{S} S-S \frac{d \mathbb{B}}{\mathbb{B}}$. Algebra completes the proof.

## 统计代写|鞅论代写Martingale Theory代考|Change of Numeraire

$$\frac{d B_t}{B_t}=0 \quad \text { and } \quad \frac{d S_t}{S_t}=\frac{d \mathbb{S}_t}{\mathbb{S}_t}-r_t d t$$

$$d\left(\frac{\mathbb{S}}{\mathbb{B}}\right)=\frac{1}{\mathbb{B}} d \mathbb{S}+\mathbb{S} d\left(\frac{1}{\mathbb{B}}\right)=\frac{d \mathbb{S}}{\mathbb{S}} \mathbb{B}-\frac{\mathbb{S}}{\mathbb{B}} \frac{d \mathbb{B}}{\mathbb{B}} .$$

$d S=\frac{d S}{S} S-S \frac{d \mathbb{B}}{1 \mathrm{~B}}$. 代数完成了证明。

## 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。

Posted on Categories:Martingale Theory, 数据科学代写, 统计代写, 统计代考, 鞅论

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## 统计代写|鞅论代写Martingale Theory代考|Essential Supremum

The following theorem (see Pham [156, p. 174]; note the proof here does not depend on $S$ being a continuous process) will be important in super- and sub-replication.
Theorem 6 (Essential Supremum) Let $X_T \geq 0$ be $\mathscr{F}T$-measurable. Let $S$ be a $\mathbb{R}^n$-valued semimartingale. Then, the cadlag modification of the process $$X_t=\operatorname{ess} \underset{\mathbb{Q} \in \mathfrak{\mathfrak { M } { l }}}{ } E^{\mathbb{Q}}\left[X_T \mid \mathscr{F}_t\right]$$
for $t \in[0, T]$ is a supermartingale for all $\mathbb{Q} \in \mathfrak{M}_l$ where $\mathfrak{M}_l=$ ${\mathbb{Q} \sim \mathbb{P}: S$ is a $\mathbb{Q}$ local martingale $}$.

## 统计代写|鞅论代写Martingale Theory代考|Optional Decomposition

The next theorem (see Follmer and Kabanov [62]) will also prove useful be in superand sub-replication.
Theorem 7 (Optional Decomposition) Let $S$ be a $\mathbb{R}^n$-valued semimartingale.
Let $\mathfrak{M}l \neq \emptyset$ where $\mathfrak{M}_l={\mathbb{Q} \sim \mathbb{P}: S$ is a $\mathbb{Q}$ local martingale $}$. Let $X$ be a local supermartingale with respect to all $\mathbb{Q} \in \mathfrak{M}_l$. Then, for any $\mathbb{Q} \in \mathfrak{M}{\text {l }}$ there exists a nondecreasing cadlag adapted process $C$ with $C_0=0$ and a predictable integrand $\alpha \in \mathscr{L}(S)$ such that
$$X_t=X_0+\int_0^t \alpha_s d S_s-C_t$$
for $t \geq 0$. If $S \geq 0$, then $\int_0^t \alpha_s d S_s$ is a local martingale with respect to $\mathbb{Q}$.

## 统计代写鞅论代写Martingale Theory代考|Essential Supremum

$$X_t=\text { ess } \underset{\mathbb{Q} \in \mathscr{M} \mathbb{I}}{ } E^{\mathbb{Q}}\left[X_T \mid \mathscr{F}_t\right]$$

X_t=X_0+\int_0^t \alpha_s d S_s-C_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。

Posted on Categories:Martingale Theory, 数据科学代写, 统计代写, 统计代考, 鞅论

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## 统计代写|鞅论代写Martingale Theory代考|Arbitrage Pricing Theory

We consider a continuous-time setting with time denoted $t \in[0, \infty)$. We are given a filtered probability space $(\Omega, \mathscr{F}, \mathbb{F}, \mathbb{P})$ where $\Omega$ is the state space with generic element $\omega \in \Omega, \mathscr{F}$ is a $\sigma$-algebra representing the set of events, $\mathbb{F}=\left(\mathscr{F}t\right){0 \leq t \leq \infty}$ is a filtration, and $\mathbb{P}$ is a probability measure defined on $\mathscr{F}$. A filtration is a collection of $\sigma$-algebras, which are increasing, i.e. $\mathscr{F}_s \subseteq \mathscr{F}_t$ for $0 \leq s \leq t \leq \infty$.

A random variable is a mapping $Y: \Omega \rightarrow \mathbb{R}$ such that $Y$ is $\mathscr{F}$-measurable, i.e. $Y^{-1}(A) \in \mathscr{F}$ for all $A \in \mathscr{B}(\mathbb{R})$ where $\mathscr{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$, i.e. the smallest $\sigma$-algebra containing all open intervals $(s, t)$ with $s \leq t$ for $s, t \in \mathbb{R}$ (see Ash [3, p. 8]).

A stochastic process is a collection of random variables indexed by time, i.e. a mapping $X:[0, \infty) \times \Omega \rightarrow \mathbb{R}$, denoted variously depending on the context, $X(t, \omega)=X(t)=X_t$. It is adapted if $X_t$ is $\mathscr{F}_t$-measurable for all $t \in[0, \infty)$.
A sample path of a stochastic process is the graph of $X(t, \omega)$ across time $t$ keeping $\omega$ fixed.

We assume that the filtered probability space satisfies the usual hypotheses. The usual hypotheses are that $\mathscr{F}0$ contains the $\mathbb{P}$ null sets of $\mathscr{F}$ and that the filtration $\mathbb{F}$ is right continuous. Right continuous means that $\mathscr{F}_t=\cap{u>t} \mathscr{F}_u$ for all $0 \leq$ $t<\infty$. Letting $\mathscr{F}_0$ contains the $\mathbb{P}$ null sets of $\mathscr{F}$ facilitates the measurability of various events, random variables, and stochastic processes. Right continuity implies the important result that given a random variable $\tau: \Omega \rightarrow[0, \infty],{\tau(\omega) \leq t} \in \mathscr{F}_t$ for all $t$ if and only if ${\tau(\omega)<t} \in \mathscr{F}_t$ for all $t$, see Protter [158, p. 3]. This fact will be important with respect to the mathematics of stopping times, which are introduced below. One can think of right continuity as implying that the information at time $t^{+}$is known at time $t$, see Medvegyev [143, p. 9].

## 统计代写|鞅论代写Martingale Theory代考|Martingales

A stochastic process $X$ is a martingale with respect to $\mathbb{F}$ if
(i) $X$ is cadlag and adapted,
(ii) $E\left[\left|X_t\right|\right]<\infty$ all $t$, and
(iii) $E\left[X_t \mid \mathscr{F}_s\right]=X_s$ a.s. for all $0 \leq s \leq t<\infty$.
It is a submartingale if (iii) is replaced by $E\left[X_t \mid \mathscr{F}_s\right] \geq X_s$ a.s. It is said to be a strict submartingale if it is a submartingale but not a martingale, i.e. the inequality is strict with positive probability for some $0 \leq s \leq t<\infty$.

It is a supermartingale if (iii) is replaced by $E\left[X_t \mid \mathscr{F}_s\right] \leq X_s$ a.s. It is said to be a strict supermartingale if it is a supermartingale but not a martingale, i.e. the inequality is strict with positive probability for some $0 \leq s \leq t<\infty$.

For the definition of an expectation and a conditional expectation, see Ash [3, Chapter 6]. Within the class of martingales, uniformly integrable martingales play an important role (see Protter [158, Theorem 13, p. 9]).

## 统计代写|鞅论代写Martingale Theory代考|Martingales

(一) $X$ 是 cadlag 和改编的，
(ii) $E\left[\left|X_t\right|\right]<\infty$ 全部 $t$, 和
(iii) $E\left[X_t \mid \mathscr{F}_s\right]=X_s$ 至于所有 $0 \leq s \leq t<\infty$.

## 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。