本章导读 本章计算布朗运动的二次变差。§6.1 定义：在二进分割上 \(Q_n=\sum_j(B_{j/2^n}-B_{(j-1)/2^n})^2\)，逐项计算 \(\mathbb E[k]=1/2^n\)、\(\mathbb E[k^2]=3/2^{2n}\)、\(\mathrm{Var}(k)=2/2^{2n}\)，得 \(\mathbb E[Q_n]=1\)、\(\mathrm{Var}(Q_n)=2/2^n\to0\)，故 \(Q_n\to1\)（概率 1）；推广到 \(Q_n(t)\to t\)。由此 Def 6.1 定义二次变差 \(\langle B\rangle_t\)，Prop 6.1：\(\langle B\rangle_t=t\)（概率 1）；Def 6.2 一般 BM \(Y_t=\mu t+\sigma B_t\) 的二次变差，Prop 6.2：\(\langle Y\rangle_t=\sigma^2 t\)。§6.2 推广到任意分割（Def 6.3 分割与网格、Def 6.4 细化、Def 6.5 分割上的二次变差、Thm 6.1：网格趋零且逐层细化时 \(Q(\Pi_n)\to1\)，概率 1）。无图。

二进分割上的逐项矩 / Term-wise moments on the dyadic partition 设 \(\{B_t\}\) 是标准布朗运动。令Let \(\{B_t\}\) be a standard Brownian motion. Let

$$Q_n=\sum_{j=1}^{2^n}\underbrace{\left(B_{\frac{j}{2^n}}-B_{\frac{j-1}{2^n}}\right)^2}_{\equiv\,k(j,n)}.$$

逐项的一阶、二阶矩与方差为The first, second moments and variance of each term are

$$\mathbb E[k(j,n)]=\mathrm{Var}\!\left(B_{\frac{j}{2^n}}-B_{\frac{j-1}{2^n}}\right)=\frac1{2^n},$$

$$\mathbb E[k(j,n)^2]=\mathbb E\!\left[\left(B_{\frac1{2^n}}\right)^4\right]\overset{\text{scaling}}{=}\frac1{2^{2n}}\,\mathbb E[(B_1)^4]=\frac3{2^{2n}},$$

$$\mathrm{Var}(k(j,n))=\mathbb E[k(j,n)^2]-\mathbb E[k(j,n)]^2=\frac3{2^{2n}}-\frac1{2^{2n}}=\frac2{2^{2n}},$$

其中用到 \(\mathbb E[(B_1)^4]=3\) 与缩放性 \(B_{1/2^n}\overset{d}{=}2^{-n/2}B_1\)。where we used \(\mathbb E[(B_1)^4]=3\) and the scaling \(B_{1/2^n}\overset{d}{=}2^{-n/2}B_1\).

\(Q_n\) 的极限 / The limit of \(Q_n\) 对 \(Q_n\) 取期望与方差：Taking the expectation and variance of \(Q_n\):

$$\mathbb E[Q_n]=\mathbb E\!\left[\sum_{j=1}^{2^n}k(j,n)\right]=2^n\,\mathbb E[k(j,n)]=2^n\cdot\frac1{2^n}=1,$$

$$\mathrm{Var}(Q_n)\overset{\text{indep.}}{=}2^n\,\mathrm{Var}(k(j,n))=2^n\cdot\frac2{2^{2n}}=\frac2{2^n}\to0\quad(n\to\infty),$$

故 \(Q_n=1\) 以概率 1 成立（当 \(n\to\infty\)）。类似地，定义 \(Q_n(t)=\sum_{j:\,j/2^n\le t}\left(B_{j/2^n}-B_{(j-1)/2^n}\right)^2\)，同样逻辑给出 \(Q_n(t)=t\) 以概率 1 成立。可换记号，把 \(Q_n(t)\) 的极限定义为二次变差。so \(Q_n=1\) with probability 1 (as \(n\to\infty\)). Similarly, defining \(Q_n(t)=\sum_{j:\,j/2^n\le t}\left(B_{j/2^n}-B_{(j-1)/2^n}\right)^2\), the same logic gives \(Q_n(t)=t\) with probability 1. We can switch notation to define the limit of \(Q_n(t)\) as the quadratic variation.

定义 6.1、命题 6.1 / Definition 6.1, Proposition 6.1 定义 6.1（二次变差）：标准布朗运动 \(\{B_t\}\) 的二次变差 \(\langle B\rangle_t\) 定义为Definition 6.1 (Quadratic Variation): the quadratic variation \(\langle B\rangle_t\) of a standard Brownian motion \(\{B_t\}\) is defined as

$$\langle B\rangle_t=\lim_{n\to\infty}\sum_{j:\,j/2^n\le t}\left(B_{\frac{j}{2^n}}-B_{\frac{j-1}{2^n}}\right)^2.$$

命题 6.1：以概率 1，\(\langle B\rangle_t=t\)。（证明见上文论证。）Proposition 6.1: with probability one, \(\langle B\rangle_t=t\). (Proof: see the argument above.)

定义 6.2、命题 6.2（一般二次变差）/ Definition 6.2, Proposition 6.2 (general) 定义 6.2（一般二次变差）：设 \(\{Y_t\}\) 是布朗运动，\(Y_t=\mu t+\sigma B_t\)（\(\forall t\)）。定义其一般二次变差 \(\langle Y\rangle_t=\lim_{n\to\infty}\sum_{j:\,j/2^n\le t}\left(Y_{j/2^n}-Y_{(j-1)/2^n}\right)^2\)。命题 6.2：以概率 1，\(\langle Y\rangle_t=\sigma^2 t\)。Definition 6.2 (More general quadratic variation): let \(\{Y_t\}\) be a Brownian motion with \(Y_t=\mu t+\sigma B_t\) (for all \(t\)). Define its general quadratic variation \(\langle Y\rangle_t=\lim_{n\to\infty}\sum_{j:\,j/2^n\le t}\left(Y_{j/2^n}-Y_{(j-1)/2^n}\right)^2\). Proposition 6.2: with probability one, \(\langle Y\rangle_t=\sigma^2 t\).

命题 6.2 证明 / Proof of Proposition 6.2 由 \(Y_t=\mu t+\sigma B_t\)，每个增量 \(Y_{j/2^n}-Y_{(j-1)/2^n}=\sigma\!\left(B_{j/2^n}-B_{(j-1)/2^n}\right)+\dfrac{\mu}{2^n}\)。展开平方并求和取极限：Since \(Y_t=\mu t+\sigma B_t\), each increment is \(Y_{j/2^n}-Y_{(j-1)/2^n}=\sigma\!\left(B_{j/2^n}-B_{(j-1)/2^n}\right)+\dfrac{\mu}{2^n}\). Expanding the square, summing and taking the limit:

$$\begin{aligned}\langle Y\rangle_t&=\sigma^2\lim_{n\to\infty}\underbrace{\sum_{j:\,j/2^n\le t}\!\left(B_{\frac{j}{2^n}}-B_{\frac{j-1}{2^n}}\right)^2}_{\to\,t}\\&\quad+\lim_{n\to\infty}\frac{2\sigma\mu}{2^n}\underbrace{\sum_{j:\,j/2^n\le t}\!\left(B_{\frac{j}{2^n}}-B_{\frac{j-1}{2^n}}\right)}_{\to\,0}\\&\quad+\lim_{n\to\infty}\frac{\mu^2}{2^{2n}}\underbrace{\sum_{j:\,j/2^n\le t}1}_{\sim\,t\cdot2^n}\\&=\sigma^2 t,\end{aligned}$$

其中主项收敛到 \(\sigma^2 t\)，交叉项与最后一项（\(\sim\mu^2 t/2^n\)）都趋于 \(0\)。\(\blacksquare\)where the main term converges to \(\sigma^2 t\), and both the cross term and the last term (\(\sim\mu^2 t/2^n\)) tend to \(0\). \(\blacksquare\)

定义 6.3–6.5（分割、细化、分割上的二次变差）/ Definitions 6.3–6.5 定义 6.3（分割）：$[0,1]$ 的一个分割是有限个时刻的集合。典型分割 \(\Pi_n:0=t_0网格 (mesh) 定义为 \(\mathrm{mesh}(\Pi_n)=\max(t_j-t_{j-1})\)。定义 6.4（细化分割）：称分割 \(\Pi_{n+1}\) 细化 \(\Pi_n\)，若 \(k_{n+1}>k_n\) 且 \(\Pi_n\) 中所有时刻都在 \(\Pi_{n+1}\) 中。定义 6.5（分割 \(\Pi_n\) 上的二次变差）：给定 \(\Pi_n\)，定义其上的二次变差 \(Q(\Pi_n)=\lim_{n\to\infty}\sum_{j=1}^{k_n}\left(B_{t_j}-B_{t_{j-1}}\right)^2\)。Definition 6.3 (Partition): a partition of $[0,1]$ is a finite collection of times. A typical partition \(\Pi_n:0=t_0mesh defined by \(\mathrm{mesh}(\Pi_n)=\max(t_j-t_{j-1})\). Definition 6.4 (Refining partition): a partition \(\Pi_{n+1}\) refines \(\Pi_n\) if \(k_{n+1}>k_n\) and all times in \(\Pi_n\) are in \(\Pi_{n+1}\). Definition 6.5 (Quadratic variation on partition \(\Pi_n\)): given \(\Pi_n\), define the quadratic variation on \(\Pi_n\) by \(Q(\Pi_n)=\lim_{n\to\infty}\sum_{j=1}^{k_n}\left(B_{t_j}-B_{t_{j-1}}\right)^2\).

定理 6.1 / Theorem 6.1 设 \(\Pi_0,\Pi_1,\dots\) 是 $[0,1]$ 的一列分割，满足 \(\mathrm{mesh}(\Pi_n)\to0\) 且 \(\forall n\)，\(\Pi_{n+1}\) 细化 \(\Pi_n\)。则以概率 1，\(Q(\Pi_n)\to1\)。（证明省略，逻辑与上文非常相似。）Let \(\Pi_0,\Pi_1,\dots\) be a sequence of partitions of $[0,1]$ such that \(\mathrm{mesh}(\Pi_n)\to0\) and for all \(n\), \(\Pi_{n+1}\) refines \(\Pi_n\). Then with probability one, \(Q(\Pi_n)\to1\). (Proof omitted; it follows very similar logic as above.)