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## 数学代写|随机微积分代写Stochastic calculus代考|Brownian motion and Wiener measure

Given the family $\left{\lambda_t: t \geq 0\right}$ associated with a Lévy system $(\mathbf{m}, C, M)$, Kolmogorov’s consistency theorem (cf. Exercise 9.1.17 in [20]) guarantees that there is a family ${X(t): t \geq 0}$ of random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with the properties that $\mathbb{P}(X(0)=0)=1$, and, for each $n \geq 1,0=t_0<t_1 \cdots<t_n$, and $\Gamma_0, \ldots, \Gamma_n \in \mathcal{B}{\mathbb{R}^N}$, $$\mathbb{P}\left(X\left(t_m\right)-X\left(t{m-1}\right) \in \Gamma_m \quad \text { for } 0 \leq j \leq n\right)=\prod_{m=1}^n \lambda_{t_m-t_{m-1}}\left(\Gamma_m\right) .$$
In fact (cf. Chapter 4 in [20]), one can always choose these random variables so that the paths $t \rightsquigarrow X(t)$ are right-continuous and have a left limit at each $t \in(0, \infty)$. That is, although they may have discontinuities, their only discontinuities are simple jumps and not oscillatory ones. Furthermore, a major goal of this section is to prove that the paths can be chosen to be continuous when $M=0$.

## 数学代写|随机微积分代写Stochastic calculus代考|L´evy’s construction of Brownian Motion

The family of measures corresponding to the Lévy system $(\mathbf{0}, \mathbf{I}, 0)$ are the Gaussian measures $\gamma_{0, t \mathbf{I}}$, and a family ${B(t): t \geq 0}$ of random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying
$$\begin{gathered} \mathbb{P}\left(B(0) \in \Gamma_0 \text { and } B\left(t_m\right)-B\left(t_{m-1}\right) \in \Gamma_m \text { for } 0 \leq j \leq n\right) \ =\mathbf{1}{\Gamma_0}(\mathbf{0}) \prod{m=1}^n \gamma_{\mathbf{0},\left(t_m-t_{m-1}\right) \mathbf{I}}\left(\Gamma_m\right) \end{gathered}$$
is called an $\mathbb{R}^N$-valued Brownian motion if, $\mathbb{P}$-almost surely, $t \rightsquigarrow B(t)$ is continuous. The first person to prove the existence of such random variables was $\mathrm{N}$. Wiener, but, because it is more transparent than Wiener’s, we will use a proof devised by P. Lévy.

To understand Lévy’s idea, it is best to begin by assuming that a Brownian motion exists and examine its polygonal approximations. Thus, suppose that ${B(t): t \geq 0}$ is a Brownian motion, and, for $n \geq 0$, let $t \rightsquigarrow B_n(t)$ be the polygonal path that linearly interpolates $t \rightsquigarrow B(t)$ between times $m 2^{-n}$. In other words, $B_n\left(m 2^{-n}\right)=B\left(m 2^{-n}\right)$ and
$$B_n(t)=\left(m+1-2^n t\right) B\left(m 2^{-n}\right)+\left(2^n t-m\right) B\left((m+1) 2^{-n}\right)$$
for $m \geq 0$ and $t \in I_{m, n}:=\left[m 2^{-n},(m+1) 2^{-n}\right]$. The distribution of each individual family $\left{B_n(t): t \geq 0\right}$ is very easy to understand, but what we need to understand is the relationship between successive families. Obviously, since $B_{n+1}\left(m 2^{-n}\right)=B_n\left(m 2^{-n}\right)$ and $t \rightsquigarrow B_{n+1}(t)-B_n(t)$ is linear on the intervals $I_{2 m-2, n+1}$ and $I_{2 m-1, n+1}$, the maximum difference between $B_{n+1}(\cdot)$ and $B_n(\cdot)$ occurs at times $(2 m-1) 2^{-n-1}$. With this in mind, set $X_{m, 0}=$ $B_0(m)-B_0(m-1)$ and, for $m \geq 1$ and $n \geq 0$,
\begin{aligned} X_{m, n+1} &=2^{\frac{n}{2}+1}\left(B_{n+1}\left((2 m-1) 2^{-n-1}\right)-B_n\left((2 m-1) 2^{-n-1}\right)\right) \ &=2^{\frac{n}{2}+1}\left(B\left((2 m-1) 2^{-n-1}\right)-\frac{B\left((m-1) 2^{-n}\right)+B\left(m 2^{-n}\right)}{2}\right) \ &=2^{\frac{n}{2}}\left[\left(B\left((2 m-1) 2^{-n-1}\right)-B\left((m-1) 2^{-n}\right)\right)\right.\ &\left.\quad-\left(B\left(m 2^{-n}\right)-B\left((2 m-1) 2^{-n-1}\right)\right)\right] \end{aligned}

# 随机微积分代考

## 数学代写|随机微积分代写Stochastic calculus代考|Brownian motion and Wiener measure

$$\mathbb{P}\left(X\left(t_m\right)-X(t m-1) \in \Gamma_m \quad \text { for } 0 \leq j \leq n\right)=\prod_{m=1}^n \lambda_{t_m-t_{m 1}}\left(\Gamma_m\right) .$$

## 数学代写|随机微积分代写Stochastic calculus代考|L´evy’s construction of Brownian Motion

$$\mathbb{P}\left(B(0) \in \Gamma_0 \text { and } B\left(t_m\right)-B\left(t_{m-1}\right) \in \Gamma_m \text { for } 0 \leq j \leq n\right)=\mathbf{1} \Gamma_0(\mathbf{0}) \prod m=1^n \gamma_{0,\left(t_m-t_{m-1}\right) \mathrm{I}}\left(\Gamma_m\right)$$

$$B_n(t)=\left(m+1-2^n t\right) B\left(m 2^{-n}\right)+\left(2^n t-m\right) B\left((m+1) 2^{-n}\right)$$

$$X_{m, n+1}=2^{\frac{n}{2}+1}\left(B_{n+1}\left((2 m-1) 2^{-n-1}\right)-B_n\left((2 m-1) 2^{-n-1}\right)\right) \quad=2^{\frac{n}{2}+1}\left(B\left((2 m-1) 2^{-n-1}\right)-\frac{B\left((m-1) 2^{-n}\right)+B\left(m 2^{-n}\right)}{2}\right)=2^{\frac{n}{2}}[(B((2 m$$

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