# Brownian 运动与二次变差

证明确定性网格 QV 的 L2 收敛和 dyadic 几乎处处收敛，再排除有限总变差。

Entry: zh-qt15 | Node: QT15 | Language: zh | Editorial revision: 2026-09-21

## Teaching instructions
你是这篇中文学习单元的教学 Agent。读者具备本包列出的先修：正态矩、L2/概率收敛、Chebyshev 与 Borel–Cantelli。
先实际取得 required_readings 中本次所选单元并读完，核对版本、页码与公式；已有同会话同版完整读取可以复用。只取得摘要或目录不得声称完成。指定原件若无法取得，先说明具体缺口；只有本包中已经具名核过等价范围的完整数学证明，才可在对应数学步骤内作为替代，并须实际读完且记录替代正文、版本与支持步骤。若本次必读仍有缺失，就停止依赖该内容的实质讲解；研究样本、训练安排、图表结果与作者主张不得以本站概述替代，也不得声称原件已经读过。runtime_reading_log 是你的实际运行记录，交付的空数组不是已读。
本篇任务：先问 n=64 到256时，首路径相对二次变差极限 1 的绝对误差从 0.020574830905 增至 0.034184022561，是否反驳收敛。让读者独立写 Gaussian 四阶矩、方差界、dyadic 求和和有限总变差反证。
先让读者尝试，再按所缺的一步解释，不将全部课文一次复述。完整证明需要标明每项条件在哪一步用到，练习给出完整解析。只采用 supplied_inputs 的本篇切片和已链接全量数据，区分教学模型、真实记录、作者论文结果。图不是证明，模拟不是现实规律；不以预测概率替换定价测度。禁止从分位数拟造分布或另抽浏览器随机数冒充冻结路径。最后问：读者只读完这个词条，真的能学明白吗？用迁移题实际判断，明确剩余能力缺口。

Before substantive teaching, actually retrieve every required reading unit for the selected scope. Read its complete designated section, including necessary assumptions, tables and footnotes. A working URL or an editorial access date is not a runtime reading receipt. Record the actual version, location, scope and what it supports. If unavailable, use a previously verified equivalent source; if the required unit remains unavailable, identify that gap rather than teach it from memory. Start runtime_reading_log empty. Once reading is complete, use a substantive diagnostic or follow the reader's request for direct explanation. Advance one complete reasoning task at a time; skip mastered basics. Distinguish original facts, supplied teaching assumptions and inference.

## Required readings and runtime protocol
```json
{
  "export_mode": "public",
  "required_readings": [
    {
      "source_id": "QGHI-LALLEY-BM",
      "access": {
        "kind": "pdf_full_text",
        "uri": "https://galton.uchicago.edu/~lalley/Courses/385/BrownianMotion.pdf"
      },
      "required_unit": {
        "locator": "§1.1 pp.1–3",
        "scope": "Wiener 定义与存在性陈述",
        "purpose": "仅采用定义"
      },
      "supports": "标准 Wiener 过程定义及存在性陈述；本稿不重证存在性。",
      "title": "Brownian Motion",
      "authors": [
        "Steven P. Lalley"
      ],
      "version": "Undated chapter PDF; no inferred revision year",
      "fallback_source_ids": []
    },
    {
      "source_id": "QGHI-LALLEY-ITO12",
      "access": {
        "kind": "pdf_full_text",
        "uri": "https://www.stat.uchicago.edu/~lalley/Courses/385/Old/ItoIntegral-2012.pdf"
      },
      "required_unit": {
        "locator": "§3.3 pp.10–12",
        "scope": "QV 与积分 WdW 完整单元",
        "purpose": "结合本篇 Gaussian 矩与 dyadic BC 证明"
      },
      "supports": "渐进可测密度、简单积分等距、延拓、停止、局部化与一维 Itô 公式。p.8 下标和 p.19 漂移步长按正确推导采用；Doob 强 L2 估计及一般系数传递在本站完整展开。",
      "title": "Notes on the Itô Calculus",
      "authors": [
        "Steven P. Lalley"
      ],
      "version": "2012-05-15",
      "fallback_source_ids": []
    }
  ],
  "optional_readings": [],
  "runtime_reading_log": [],
  "supplied_inputs": {
    "competence": "正态矩、L2/概率收敛、Chebyshev 与 Borel–Cantelli。",
    "static_equivalent": "本篇 reader 全部默认表、证明及题解；HTML 禁用脚本仍可读。",
    "attachments": [
      {
        "title": "本篇完整静态阅读、全部题解与证明",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/static/QT15.html",
        "kind": "html"
      },
      {
        "title": "本篇同源 Markdown",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/static/QT15.md",
        "kind": "markdown"
      },
      {
        "title": "common-first-path.csv",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/common-first-path.csv",
        "kind": "csv",
        "bytes": 23281
      },
      {
        "title": "reproduce.py",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/compute/reproduce.py",
        "kind": "py",
        "bytes": 10867
      },
      {
        "title": "shared-normal-array.npz",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/shared-normal-array.npz",
        "kind": "npz",
        "bytes": 16118323
      },
      {
        "title": "完整实际结果及指定单元",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/results.json",
        "kind": "json",
        "json_pointers": [
          "/diffusion"
        ]
      },
      {
        "title": "四个实验的唯一冻结定义",
        "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/qt-ghi-shared-experiments.json",
        "kind": "json",
        "json_pointer": "/experiments/2",
        "experiment_id": "EXP-QT15-17-DIFFUSION-01"
      }
    ],
    "frozen": {
      "rng": {
        "library": "NumPy 2.4.6",
        "generator": "Generator(PCG64(15161724))",
        "draw": "standard_normal((8192,256))",
        "seed": 15161724,
        "shape": [
          8192,
          256
        ],
        "draw_order": "Exactly as the existing source algorithm; no preceding draws."
      },
      "brownian_horizon": 1,
      "finest_intervals": 256,
      "nested_intervals": [
        4,
        16,
        64,
        256
      ],
      "gbm_P_illustration": {
        "S0": 100,
        "mu_P": 0.08,
        "sigma": 0.2,
        "T_years": 1,
        "identity": "Teaching physical model; not fitted to observed returns."
      }
    },
    "default_outputs": {
      "grids": [
        {
          "steps": 4,
          "first_qv": 0.7944594147350206,
          "mean_qv": 0.9906292333607486,
          "sample_variance": 0.506927228120233,
          "theory_variance": 0.5,
          "left": -0.19728199481057607,
          "right": 0.5971774199244447,
          "integral": -0.3000522874430651,
          "integral_mse": 0.12673828965562223,
          "theory_mse": 0.125,
          "euler_rmse": 1.6218905290221228,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.865174681370263e-14,
          "right_left_max_residual": 4.440892098500626e-15
        },
        {
          "steps": 16,
          "first_qv": 1.0792151299895332,
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          "left": -0.33965985243783225,
          "right": 0.7395552775517009,
          "integral": -0.3000522874430651,
          "integral_mse": 0.031220588751377322,
          "theory_mse": 0.03125,
          "euler_rmse": 0.7879970305170926,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.7763568394002505e-14,
          "right_left_max_residual": 1.7763568394002505e-15
        },
        {
          "steps": 64,
          "first_qv": 1.0205748309050482,
          "mean_qv": 0.9979784834959156,
          "sample_variance": 0.031275134203983666,
          "theory_variance": 0.03125,
          "left": -0.31033970289559004,
          "right": 0.7102351280094583,
          "integral": -0.3000522874430651,
          "integral_mse": 0.007818850741888548,
          "theory_mse": 0.0078125,
          "euler_rmse": 0.3906616566182395,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.687538997430238e-14,
          "right_left_max_residual": 1.9984014443252818e-15
        },
        {
          "steps": 256,
          "first_qv": 0.9658159774394378,
          "mean_qv": 1.0007154342405915,
          "sample_variance": 0.007928244277278857,
          "theory_variance": 0.0078125,
          "left": -0.2829602761627848,
          "right": 0.6828557012766532,
          "integral": -0.3000522874430651,
          "integral_mse": 0.001981947080043741,
          "theory_mse": 0.001953125,
          "euler_rmse": 0.19619282035111724,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.7763568394002505e-14,
          "right_left_max_residual": 1.7763568394002505e-15
        }
      ],
      "terminal": {
        "n": 8192,
        "mean": 108.19699073155812,
        "sample_sd": 21.797741228015408,
        "mean_mc_se": 0.24083329120123867,
        "mse_to_zero": 12181.672325334242,
        "mse_mc_se": 55.97140472475497,
        "q05": 76.595919234743,
        "q95": 147.22698979604334
      },
      "theory_mean": 108.32870676749586,
      "log_drift": 0.06
    }
  }
}
```

## Supplied entry
<a id="qt15-object"></a>
## 从增量尺度开始，而不是从一条锯齿线开始

看一条模拟曲线，最多能提出猜想；要知道链式法则为什么多出一项，必须计算小增量的平方总和。本篇会完整证明确定性分割上的 $L^2$ 二次变差收敛，再得到 dyadic 分割上的几乎必然收敛，并反证 Brownian 路径具有有限总变差。先修是高斯分布的矩、方差、概率收敛和基本测度语言。

令 $(\Omega,\mathcal F,(\mathcal F_t),P)$ 为过滤概率空间。相对于此过滤的标准 Brownian 运动 $W$ 满足：$W_0=0$，过程适应且有连续路径，并且对 $s<t$，$W_t-W_s$ 独立于 $\mathcal F_s$，服从 $N(0,t-s)$。我们采用连续版本。存在性是另一个定理，本篇不以有限网格模拟代替它。[^bm]

“相对于过滤”不能省略。如果把未来的 $W_T$ 塞进 $\mathcal F_0$，路径和无条件分布都可以不变，但未来增量不再独立于当前信息。后面使用的条件零均值会失效。

<a id="qt15-moments"></a>
## 平方为什么留下一个非零总量

设 $Z\sim N(0,1)$，密度为 $\varphi$，有 $\varphi'(z)=-z\varphi(z)$。分部积分的边界项由高斯尾部消失，得到

$$
\begin{gathered}
E[Z^2]=-\int z\varphi'(z)\,dz=1,\\
E[Z^4]=-\int z^3\varphi'(z)\,dz=3.
\end{gathered}
$$

所以若 $\Delta W\sim N(0,\Delta t)$，

$$
E[(\Delta W)^2]=\Delta t,
\qquad
\operatorname{Var}((\Delta W)^2)=2(\Delta t)^2.
$$

现在固定 $T>0$ 和一个确定性分割 $\pi:0=t_0<\cdots<t_n=T$，记

$$
Q_\pi=\sum_{j=0}^{n-1}(W_{t_{j+1}}-W_{t_j})^2,
\qquad |\pi|=\max_j(t_{j+1}-t_j).
$$

不交区间增量独立，所以平方也独立，故

$$
E[Q_\pi]=T,
\qquad
E[(Q_\pi-T)^2]
=2\sum_j(\Delta t_j)^2
\le2T|\pi|.
$$

因此，对任意网格宽度趋于零的确定性分割序列，$Q_\pi\to T$ 于 $L^2$，从而也依概率收敛。这是一个完整的误差界，不需要把不同层网格的增量误认成相互独立。[^ito-qv]

<a id="qt15-as"></a>
## 从均方收敛走到路径结论

若采用 dyadic 网格，$\Delta t=T/2^m$，则对任意 $\varepsilon>0$，Chebyshev 给

$$
P(|Q_m-T|>\varepsilon)
\le\frac{2T^2}{\varepsilon^2 2^m}.
$$

右边对 $m$ 可求和。因此 Borel–Cantelli 的第一部分说明：固定 $\varepsilon$ 的坏事件只发生有限次。再取可数个 $\varepsilon=1,1/2,1/3,\ldots$ 的共同满测度集合，得到 $Q_m\to T$ 几乎必然。这里只用可求和，不要求不同 $m$ 的误差独立。

注意量词：我们固定了一组 dyadic 分割。没有证明“对所有随机、甚至依赖整条路径的分割，同时几乎必然收敛”。把确定性分割的 $L^2$ 结论与某个固定细化序列的路径结论分开，后面才不会滥用二次变差。

<a id="qt15-variation"></a>
## 连续但没有有限总变差

对连续函数 $w$，总变差是

$$
\operatorname{TV}_{[0,T]}(w)
=\sup_\pi\sum_j|w(t_{j+1})-w(t_j)|.
$$

若它有限，则每个分割都有

$$
\sum_j(\Delta w_j)^2
\le\max_j|\Delta w_j|\,
\operatorname{TV}_{[0,T]}(w).
$$

连续函数在紧区间上一致连续，故网格趋细时右边趋于零。然而 Brownian 路径在刚才的共同满测度集合上，dyadic 平方和趋于 $T>0$，矛盾。因此在任意预先固定的非退化区间上，Brownian 路径几乎必然不是有限总变差路径。

这说明不能直接把它当作普通有限变差积分的积分器。它不等于“任何形式的路径积分都不可能”，也没有顺便证明处处不可导。我们只使用刚刚建立的结论，为下一篇的均方积分构造留下明确动机。

<a id="qt15-experiment"></a>
## 同一路径、四层网格

实验固定 PCG64(15161724)，一次生成 $8192\times256$ 个标准正态数。最细增量为 $Z_j/\sqrt{256}$，较粗的 4、16、64 步由连续最细增量相加；不是每次切换都另抽一条路径。模型时钟为一年，Brownian 的平方量纲对应时间。

| 网格区间数 | 同一首路径 QV | 8,192 条路径 QV 均值 | 理论方差 $2/n$ |
| --- | --- | --- | --- |
| 4 | 0.794459415 | 0.990629233 | 0.500000000 |
| 16 | 1.079215130 | 0.997213464 | 0.125000000 |
| 64 | 1.020574831 | 0.997978483 | 0.031250000 |
| 256 | 0.965815977 | 1.000715434 | 0.007812500 |

<div data-experiment-slot="EXP-QT15-17-DIFFUSION-01"></div>

单条示例路径的平方和不是单调靠近 1：16 步时在 1 上方，256 步时又在下方。收敛不要求每一次细化都改善误差。样本均值更接近 1，也不意味着我们已经从有限数据证明了路径定理；定理来自高斯矩和上面的概率论论证。

在真实日线资料中可以计算 $\sum(\Delta\log P)^2$，但它带着具体日频、复权和缺失处理。它不是获得了价格的全部连续路径，也不能只凭数值相近就断言价格服从 GBM。这里的主实验不把事后调整价当作历史成交价。

<a id="qt15-exercise"></a>
## 迁移：加入漂移以后平方和怎样变化？

**题目。** 对常数 $a,b$，令 $X_t=at+bW_t$。沿同一组 dyadic 分割，证明平方增量和趋于 $b^2T$，并指出为什么只观察平方和不能恢复漂移 $a$。再解释：改看 $W_{2t}$，其 $[0,1]$ 平方和目标为什么不是 1。

**解析。** 展开平方：

$$
\sum(\Delta X)^2
=a^2\sum(\Delta t)^2
+2ab\sum\Delta t\,\Delta W
+b^2Q_\pi.
$$

第一项不超过 $a^2T|\pi|$。混合项的绝对值不超过

$$
2|ab|\left(\sum(\Delta t)^2\right)^{1/2}
Q_\pi^{1/2},
$$

在 dyadic 的满测度收敛集合上趋于零，第三项趋于 $b^2T$。故漂移不进入该极限，并不能由这个量识别 $a$。对 $W_{2t}$，每个时长 $\Delta t$ 的增量方差为 $2\Delta t$，相当于系数 $b=\sqrt2$ 的 Brownian 运动，因此目标为 2。时间尺度也是模型的一部分。

[^bm]: Steven P. Lalley，[Brownian Motion](https://galton.uchicago.edu/~lalley/Courses/385/BrownianMotion.pdf)，未标修订日期，§1.1，物理 pp.1–3 的定义与存在性陈述。
[^ito-qv]: Steven P. Lalley，[Notes on the Itô Calculus](https://www.stat.uchicago.edu/~lalley/Courses/385/Old/ItoIntegral-2012.pdf)，2012-05-15，§3.3，pp.10–12。本文把高斯矩、$L^2$ 误差界及 dyadic 的 Borel–Cantelli 步骤完整展开。


## Experiment inputs and static equivalents
```json
[
  {
    "id": "EXP-QT15-17-DIFFUSION-01",
    "title": "同一路径的二次变差",
    "anchor": "qt15-experiment",
    "description": "共享输入的本篇视图；保持 EXP-QT15-17-DIFFUSION-01 唯一冻结身份。",
    "inputs": {
      "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/learning-inputs.json",
      "json_pointer": "/QT15",
      "scope": "本篇 supplied_inputs 的完整同源切片；冻结参数和全部结果附件在该切片内定位。"
    },
    "outputs": {
      "grids": [
        {
          "steps": 4,
          "first_qv": 0.7944594147350206,
          "mean_qv": 0.9906292333607486,
          "sample_variance": 0.506927228120233,
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          "integral": -0.3000522874430651,
          "integral_mse": 0.12673828965562223,
          "theory_mse": 0.125,
          "euler_rmse": 1.6218905290221228,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.865174681370263e-14,
          "right_left_max_residual": 4.440892098500626e-15
        },
        {
          "steps": 16,
          "first_qv": 1.0792151299895332,
          "mean_qv": 0.9972134638881094,
          "sample_variance": 0.12488983556326173,
          "theory_variance": 0.125,
          "left": -0.33965985243783225,
          "right": 0.7395552775517009,
          "integral": -0.3000522874430651,
          "integral_mse": 0.031220588751377322,
          "theory_mse": 0.03125,
          "euler_rmse": 0.7879970305170926,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.7763568394002505e-14,
          "right_left_max_residual": 1.7763568394002505e-15
        },
        {
          "steps": 64,
          "first_qv": 1.0205748309050482,
          "mean_qv": 0.9979784834959156,
          "sample_variance": 0.031275134203983666,
          "theory_variance": 0.03125,
          "left": -0.31033970289559004,
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          "integral": -0.3000522874430651,
          "integral_mse": 0.007818850741888548,
          "theory_mse": 0.0078125,
          "euler_rmse": 0.3906616566182395,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.687538997430238e-14,
          "right_left_max_residual": 1.9984014443252818e-15
        },
        {
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          "integral": -0.3000522874430651,
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          "euler_rmse": 0.19619282035111724,
          "euler_nonpositive": 0,
          "telescoping_max_residual": 1.7763568394002505e-14,
          "right_left_max_residual": 1.7763568394002505e-15
        }
      ],
      "terminal": {
        "n": 8192,
        "mean": 108.19699073155812,
        "sample_sd": 21.797741228015408,
        "mean_mc_se": 0.24083329120123867,
        "mse_to_zero": 12181.672325334242,
        "mse_mc_se": 55.97140472475497,
        "q05": 76.595919234743,
        "q95": 147.22698979604334
      },
      "theory_mean": 108.32870676749586,
      "log_drift": 0.06
    },
    "source_experiment_id": "EXP-QT15-17-DIFFUSION-01",
    "static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/static/QT15.html#qt15-experiment",
    "result_attachment": {
      "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-ghi/data/results.json",
      "json_pointers": [
        "/diffusion"
      ]
    }
  }
]
```

## Sources
- [Brownian Motion](https://galton.uchicago.edu/~lalley/Courses/385/BrownianMotion.pdf): 标准 Wiener 过程定义及存在性陈述；本稿不重证存在性。
- [Notes on the Itô Calculus](https://www.stat.uchicago.edu/~lalley/Courses/385/Old/ItoIntegral-2012.pdf): 渐进可测密度、简单积分等距、延拓、停止、局部化与一维 Itô 公式。p.8 下标和 p.19 漂移步长按正确推导采用；Doob 强 L2 估计及一般系数传递在本站完整展开。

## Content relations
```json
[
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    "relation": "part_of",
    "to": "quant-processes",
    "reason": "主要 topic 归属"
  },
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    "relation": "requires",
    "to": "zh-qt12",
    "reason": "该段实际使用所列局部能力。",
    "required_competence": "L2/概率收敛与 Chebyshev"
  },
  {
    "from": "zh-qt15",
    "relation": "supported_by",
    "to": "QGHI-LALLEY-BM",
    "reason": "标准 Wiener 过程定义及存在性陈述；本稿不重证存在性。",
    "locator": "§1.1 pp.1–3",
    "scope": "Wiener 定义与存在性陈述"
  },
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    "from": "zh-qt15",
    "relation": "supported_by",
    "to": "QGHI-LALLEY-ITO12",
    "reason": "渐进可测密度、简单积分等距、延拓、停止、局部化与一维 Itô 公式。p.8 下标和 p.19 漂移步长按正确推导采用；Doob 强 L2 估计及一般系数传递在本站完整展开。",
    "locator": "§3.3 pp.10–12",
    "scope": "QV 与积分 WdW 完整单元"
  },
  {
    "from": "qt15-experiment",
    "relation": "illustrated_by",
    "to": "EXP-QT15-17-DIFFUSION-01",
    "reason": "唯一冻结输入在本篇的对应视图。"
  },
  {
    "from": "qt15-object",
    "relation": "supported_by",
    "to": "QGHI-LALLEY-BM",
    "reason": "对应本段作者已著明脚注；教学模型及本站补全证明的身份沿原脚注保留。",
    "locator": "Steven P. Lalley，[Brownian Motion](https://galton.uchicago.edu/~lalley/Courses/385/BrownianMotion.pdf)，未标修订日期，§1.1，物理 pp.1–3 的定义与存在性陈述。",
    "scope": "只支持此处脚注具名的定义、条件、证明或原表单元。",
    "citation_labels": [
      "bm"
    ]
  },
  {
    "from": "qt15-moments",
    "relation": "supported_by",
    "to": "QGHI-LALLEY-ITO12",
    "reason": "对应本段作者已著明脚注；教学模型及本站补全证明的身份沿原脚注保留。",
    "locator": "Steven P. Lalley，[Notes on the Itô Calculus](https://www.stat.uchicago.edu/~lalley/Courses/385/Old/ItoIntegral-2012.pdf)，2012-05-15，§3.3，pp.10–12。本文把高斯矩、$L^2$ 误差界及 dyadic 的 Borel–Cantelli 步骤完整展开。",
    "scope": "只支持此处脚注具名的定义、条件、证明或原表单元。",
    "citation_labels": [
      "ito-qv"
    ]
  }
]
```

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