### 金融代写|金融数学代写Financial Mathematics代考|ACTL20001

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融数学代写Financial Mathematics代考|Stochastic processes

The main objective of this book is to study certain families of stochastic (or random) processes in discrete time. There are two ways of seeing such objects:

• as a sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ of real random variables;
• as a single random variable $X$ taking values in the set of real sequences.
The index $n$ represents time. Since $n \in \mathbb{N}$, we speak of processes in discrete time. In the rest of this book, unless indicated otherwise, we will only consider processes taking discrete real values. The notation $E$ thus denotes a finite or countable subset of $\mathbb{R}$ and $\mathcal{E}=\mathcal{P}(E)$, the set of subsets of $E$.

DEFINITION 1.18.-A stochastic process is a sequence $X=\left(X_{n}\right)_{\mathrm{n} \in \mathbb{N}}$ of random variables taking values in $(E, \mathcal{E})$. The process $X$ is then a random variable taking values in $\left(E^{\mathbb{N}}, \mathcal{E}^{\otimes} \mathrm{N}\right)$.

EXAMPLE 1.22.- A coin is tossed an infinite number of times. This experiment is modeled by $\Omega={T, H}^{\mathbb{N}^{}}$. For $n \in \mathbb{N}^{}$, consider the mappings $X_{n}$ to $\Omega$ in $\mathbb{R}$ defined by
$$X_{n}\left(\omega_{1}, \omega_{2}, \ldots, \omega_{n}, \ldots\right)=\mathbb{1}{{T}}\left(\omega{n}\right),$$
the number of tails at the nth toss. Therefore, $X_{n}, n \in \mathbb{N}^{*}$ are discrete, real random variables and the sequence $X=\left(X_{n}\right)_{n \in \mathbb{N}}$ is a stochastic process.

DEfINITION 1.19.- Let $X=\left(X_{n}\right){n \in \mathbb{N}}$ be a stochastic process. For all $n \in \mathbb{N}$, the distribution of the vector $\left(X{0}, X_{1}, \ldots, X_{n}\right)$ is denoted by $\mu_{n}$. The probability distributions $\left(\mu_{n}\right){n \in \mathbb{N}}$ are called finite-dimensional distributions or finite-dimensional marginal distributions of the process $X=\left(X{n}\right)_{n \in \mathbb{N}}$.

PROPOSITION 1.10. – Let $X=\left(X_{n}\right){n \in \mathbb{N}}$ be a stochastic process and let $\left(\mu{n}\right){n \in \mathbb{N}}$ be its finite-dimensional distributions. Then, for all $n \in \mathbb{N}^{*}$ and $\left(A{0}, \ldots, A_{n-1}\right) \in \mathcal{E}^{n}$, we have
$$\mu_{n-1}\left(A_{0} \times \ldots \times A_{n-1}\right)=\mu_{n}\left(A_{0} \times \ldots \times A_{n-1} \times E\right)$$
In other words, the restriction of the marginal distribution of the vector $\left(X_{0}, \ldots, X_{n}\right)$ to its first $n$ coordinates is exactly the distribution of the vector $\left(X_{0}, \ldots, X_{n-1}\right)$
PROOF.- This proof directly follows from the definition of the objects. We have
\begin{aligned} \mu_{n-1}\left(A_{0} \times \ldots \times A_{n-1}\right) &=\mathbb{P}\left(X_{0} \in A_{0}, \ldots, X_{n-1} \in A_{n-1}\right) \ &=\mathbb{P}\left(X_{0} \in A_{0}, \ldots, X_{n-1} \in A_{n-1}, X_{n} \in E\right) \ &=\mu_{n}\left(A_{0} \times \ldots \times A_{n-1} \times E\right), \end{aligned}
and hence, the desired equality.
Indeed, this property completely characterizes the distribution of the process $X$ according to the following theorem.

THEOREM $1.4$ (Kolmogorov).-The canonical space $(\Omega, \mathcal{F})$ is defined in the following manner. Let $\Omega=E^{\mathbb{N}}$. The coordinate mappings $\left(X_{n}\right){n \in \mathbb{N}}$ are defined by $X{n}(\omega)=\omega_{n}$ for any $\omega=\left(\omega_{n}\right){n \in \mathbb{N}} \in \Omega$ and we write $\mathcal{F}=\sigma\left(X{n}, n \in \mathbb{N}\right)$. Let $\left(\mu_{n}\right){n \in \mathbb{N}}$ be a family of probability distributions such that 1) for any $n \in \mathbb{N}, \mu{n}$ is defined on $\left(E^{n+1}, \mathcal{E}^{\otimes(n+1)}\right)$,
2) for any $n \in \mathbb{N}^{*}$ and $\left(A_{0}, \ldots, A_{n-1}\right) \in \mathcal{E}^{n}$, we have $\mu_{n-1}\left(A_{0} \times \ldots \times A_{n-1}\right)=$ $\mu_{n}\left(A_{0} \times \ldots \times A_{n-1} \times E\right)$.

## 金融代写|金融数学代写Financial Mathematics代考|Conditional probability with respect to an event

Conditional probability accounts for the information brought in by the occurrence of one event on the probability of the occurrence of another event.

DEFINITION 2.1.-Let $B \in \mathcal{F}$ such that $\mathbb{P}(B)>0$. The conditional probability of the event $A$ given the event $B$ is the number
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$

EXAMPLE 2.1.- There are two roads to go from city $A$ to city $B$ and two roads from city $B$ to city $C$. Each of these roads is blocked by snow, with a probability $p$, independently of the others. We wish to find the probability that there is a practicable road from $A$ to $B$ given that it is not possible to travel between $A$ and $C$. We introduce the following events
$A B$ : at least one road is open between $A$ and $B$,
$B C$ : at least one road is open between $B$ and $C$.
We have $\mathbb{P}(A B)=\mathbb{P}(B C)=1-p^{2}$ since the two possible roads are blocked independently of one another. Further, the events $A B$ and $B C$ are independent. We wish to find $\mathbb{P}\left(A B \mid(A B \cap B C)^{\mathrm{c}}\right)$. We have, using independence,
$$\mathbb{P}\left(A B \mid(A B \cap B C)^{c}\right)=\frac{\mathbb{P}\left(A B \cap(A B \cap B C)^{c}\right)}{\mathbb{P}\left((A B \cap B C)^{c}\right)}$$
\begin{aligned} &=\frac{\mathbb{P}\left(A B \cap\left(A B^{c} \cup B C^{c}\right)\right)}{1-\mathbb{P}(A B \cap B C)} \ &=\frac{\mathbb{P}\left(\left(A B \cap A B^{c}\right) \cup\left(A B \cap B C^{c}\right)\right)}{1-\mathbb{P}(A B) \mathbb{P}(B C)} \end{aligned}
\begin{aligned} &=\frac{\mathbb{P}\left(A B \cap B C^{c}\right)}{1-\mathbb{P}(A B) \mathbb{P}(B C)} \ &=\frac{\mathbb{P}(A B) \mathbb{P}\left(B C^{c}\right)}{1-\mathbb{P}(A B) \mathbb{P}(B C)} \ &=\frac{\left(1-p^{2}\right) p^{2}}{1-\left(1-p^{2}\right)^{2}} \end{aligned}
The probability that there is a practicable route from $A$ to $B$ given that we cannot travel between $A$ and $C$ is thus $\frac{\left(1-p^{2}\right) p^{2}}{1-\left(1-p^{2}\right)^{2}}$.

PROPOSITION 2.1.- Let $B \in \mathcal{F}$ such that $\mathbb{P}(B)>0$. The mapping $\mathbb{P}(\cdot \mid B)$ from $\mathcal{F}$ to $\mathbb{R}^{+}$defined by
$$A \longmapsto \mathbb{P}(A \mid B)$$
is a probability over $(\Omega, \mathcal{F})$. It is called the conditional distribution given $B$.

## 金融代写|金融数学代写Financial Mathematics代考|Conditional expectation

The theorem given below broadens the definition of conditional expectation when we condition with respect to a $\sigma$-algebra.

THEOREM 2.1.-Let $X$ be a random variable in $L^{1}(\Omega, \mathcal{F}, \mathbb{P})$ and let $\mathcal{G}$ be a sub- $\sigma$-algebra of $\mathcal{F}$. Then, there exists a unique random variable $Y \in L^{1}(\Omega, \mathcal{F}, \mathbb{P})$ such that
1) $Y$ is $\mathcal{G}$-measurable;
2) for any $A \in \mathcal{G}$, we have $\mathbb{E}\left[X \mathbb{1}{A}\right]=\mathbb{E}\left[Y \mathbb{1}{A}\right]$.

The unicity is intended almost surely, that is, if two random variables $Y$ and $Z$ satisfy the above properties, then $\mathbb{P}(Y=Z)=1$. Informally, $Y$ is a random, $\mathcal{G}$-measurable variable, which resembles $X$ in the sense that it coincides with $X$ on $\mathcal{G}$. In a sense, which will be explained in detail in section $2.3$, it is the best approximation of $X$ that is $\mathcal{G}$-measurable. In other words, $Y$ is the best prediction of $X$ when the quantity of information revealed by the $\sigma$-algebra $\mathcal{G}$ is available.
PROOF. – Existence and unicity are demonstrated separately.
Existence: first case: assume that $X \geq 0$. We apply the Radon-Nykodym theorem 1 with $\mu=\mathbb{P}$ which is a measure on the probability space $(\Omega, \mathcal{G})$ since $\mathcal{G}$ is a sub- $\sigma$-algebra of $\mathcal{F}$. As regards $\nu$, consider the measure defined over $(\Omega, \mathcal{G})$ by:
$$\forall B \in \mathcal{G}, \nu(B)=\mathbb{E}\left[X \mathbb{1}{B}\right]=\int{B} X d \mathbb{P}$$
We then have $\nu \ll \mu$. Indeed, we have
$$\mathbb{P}(A)=0 \Longrightarrow \mathbb{1}{A}=0 \mathbb{P}-a . s . \Longrightarrow X \mathbb{1}{A}=0 \mathbb{P}-\text { a.s. } \Longrightarrow \nu(A)=0 .$$
Thus, using the Radon-Nykodym theorem, there exists a function $Y$ that is $\mathcal{G}$-measurable (i.e. a random $\mathcal{G}$-measurable variable $Y$ ), such that for any $A \in \mathcal{G}$, we have $\nu(A)=\int_{A} Y d \mathbb{P}$, which translates into the desired equality $\mathbb{E}\left[X \mathbb{1}{A}\right]=\mathbb{E}\left[Y \mathbb{1}{A}\right]$.
Second case: if the random variable $X$ has any sign, it is decomposed as the difference of its positive part and its negative part: $X=X^{+}-X^{-}$, where
$$X^{+}=\max (X, 0) \text { and } X^{-}=\max (-X, 0)$$

## 金融代写|金融数学代写Financial Mathematics代考|Stochastic processes

• 作为一个序列(Xn)n∈ñ真实随机变量；
• 作为单个随机变量X取实数序列集中的值。
指数n代表时间。自从n∈ñ，我们说的是离散时间的过程。在本书的其余部分，除非另有说明，否则我们将只考虑采用离散实数值的过程。符号和因此表示一个有限的或可数的子集R和和=磷(和)，子集的集合和.

Xn(ω1,ω2,…,ωn,…)=1吨(ωn),

μn−1(一个0×…×一个n−1)=μn(一个0×…×一个n−1×和)

μn−1(一个0×…×一个n−1)=磷(X0∈一个0,…,Xn−1∈一个n−1) =磷(X0∈一个0,…,Xn−1∈一个n−1,Xn∈和) =μn(一个0×…×一个n−1×和),

2) 对于任何n∈ñ∗和(一个0,…,一个n−1)∈和n， 我们有μn−1(一个0×…×一个n−1)= μn(一个0×…×一个n−1×和).

## 金融代写|金融数学代写Financial Mathematics代考|Conditional probability with respect to an event

=磷(一个乙∩(一个乙C∪乙CC))1−磷(一个乙∩乙C) =磷((一个乙∩一个乙C)∪(一个乙∩乙CC))1−磷(一个乙)磷(乙C)

=磷(一个乙∩乙CC)1−磷(一个乙)磷(乙C) =磷(一个乙)磷(乙CC)1−磷(一个乙)磷(乙C) =(1−p2)p21−(1−p2)2

## 金融代写|金融数学代写Financial Mathematics代考|Conditional expectation

1)是是G- 可测量的；
2) 对于任何一个∈G, 我们有 $\mathbb{E}\left[X \mathbb{1} {A}\right]=\mathbb{E}\left[Y \mathbb{1} {A}\right]$。

∀乙∈G,ν(乙)=和[X1乙]=∫乙Xd磷

X+=最大限度(X,0) 和 X−=最大限度(−X,0)

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## MATLAB代写

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