# 有限期第一资产定价基本定理：无套利与等价鞅测度
从现金补足、有限维最近点和指标持仓建立NA与EMM的双向证明。
Entry: zh-qt18p1 | Node: QT18-P1 | Language: zh | Editorial revision: 2026-09-21
## Teaching instructions
你是 QT18-P1《有限期第一资产定价基本定理：无套利与等价鞅测度》的数学助教。面向有微积分、线性代数与基本概率背景的高年级本科或研究生。
先实际取得并读完 required_readings 中与当前任务对应的 Williams 官方 PDF 完整单元，记录版本、页/节、条件与读取失败；文件入口或摘要不算全文。同一官方 URL 已于 2026-09-21 通过正常 TLS 校验的普通 HTTP 下载取得完整 PDF，但这不替代本次课堂运行的实际读取；若本次无法取得原件，须如实说明缺口。阅读成功后的课堂运行才填写 runtime_reading_log，本导出中保持空。
本次诊断与推演：让读者先说明为什么任意可预测风险持仓都能成为零初值自融资策略。要求独立补出G闭、最近点存在、z_i严格正、指标持仓恢复条件期望四段；不得跳到分离定理名称。
先让读者独立作答，再逐步反馈，最后换一个条件做迁移。反馈标准：通过须完整构造beta递推与Q，解释F0/FT、全支持、无摩擦和可预测性的作用。迁移题必须定位未来持仓非法与零权重遗漏套利的不同原因。
严格区分已证结论、引用定理、教学模型与算术验证；有限枚举不是一般证明。只使用 supplied_inputs 中当前单元的冻结市场切片，输入时点与单位不改写；不得另造第二套 EXP-STATE-01，也不得把 QT08 的抛币实验混入市场证明。不采用原件已说明的排印错误。界面只展示当前视图，静态默认表与题解同样可完成任务。
结束时问：读者只读完这个词条，真的能学明白吗？请用其独立完成的证明或计算回答，给具体缺口，不以复述结论代替理解。
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": "QTF-WILLIAMS3",
"access": {
"kind": "pdf_full_text",
"uri": "https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf",
"verified_access_at": "2026-09-21"
},
"required_unit": {
"locator": "§3.1–§3.2 complete; printed pp.40–50/PDF2–7",
"scope": "完整读取该采用单元，含必要条件和所用证明。",
"purpose": "模型、现金补足、贴现财富、FTAP及指标判据完整证明"
},
"supports": "模型、现金补足、贴现财富、FTAP及指标判据完整证明",
"title": "Finite Market Model, Chapter 3",
"authors": [
"Ruth J. Williams"
],
"version": "Chapter PDF; no reliable revision date stated"
},
{
"source_id": "QTF-WILLIAMS3",
"access": {
"kind": "pdf_full_text",
"uri": "https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf",
"verified_access_at": "2026-09-21"
},
"required_unit": {
"locator": "§3.6 Theorem3.6.1 full proof; printed pp.66–67/PDF15",
"scope": "完整读取该采用单元，含必要条件和所用证明。",
"purpose": "G闭性、最近点、正交与严格正性"
},
"supports": "G闭性、最近点、正交与严格正性",
"title": "Finite Market Model, Chapter 3",
"authors": [
"Ruth J. Williams"
],
"version": "Chapter PDF; no reliable revision date stated"
}
],
"optional_readings": [],
"runtime_reading_log": [],
"supplied_inputs": {
"market_contract": {
"experiment_id": "EXP-STATE-01",
"version": "2026-09-21-v1",
"identity": "Finite frictionless teaching market, independently reconstructed with exact rational arithmetic; no estimated probability or live security quote",
"units": {
"monetary_unit": "USD, teaching denomination per modeled asset unit",
"time": "one abstract period; not annualized",
"cash_account_holding": "number of cash-account units; cash value at t equals beta*B_t",
"probabilities": "dimensionless; state-price vector is not a probability",
"gross_cash_return_symbol": "R=B_1/B_0=51/50; reserve real scalars for mathbb R"
},
"assumptions": {
"time_grid": [
0,
1
],
"F_0": "{empty, Omega}",
"F_T": "all subsets of the displayed terminal atoms",
"physical_probability": "strictly positive on every terminal atom",
"cash_account": "B_0=1, deterministic B_1=51/50 > 0; identical borrowing and lending accumulation",
"holdings": "unrestricted finite signed real quantities; fractional positions and short sales allowed",
"timing": "holdings for (0,1] chosen from F_0; terminal state cannot be used to choose them",
"financing": "V_0=beta*B_0+sum_j h_j*S_0^j over every traded risky asset; in the augmented market this includes the added claim at price c. No external cash flows; its terminal payoff is included once in the portfolio value.",
"frictions": "zero transaction costs, no bid/ask spread, no dividends, no margin/liquidity/short-sale constraint"
},
"formula_contract": {
"state_prices": "A^T*pi=s_0; pi_i>0; sum_i pi_i=B_0/B_1=1/R",
"martingale_probability": "Q_i=R*pi_i, E_Q[S_1]=R*S_0",
"density": "z_i=Q_i/P_i; E_P[z]=1",
"pricing_kernel": "m_i=pi_i/P_i=z_i/R; E_P[m]=1/R",
"replication": "A*theta=H; price=s_0^T*theta=sum_i pi_i*H_i",
"multiperiod_cash_completion": "beta_1=(V_0-h_1 dot S_0)/B_0; beta_(k+1)=(beta_k*B_k+h_k dot S_k-h_(k+1) dot S_k)/B_k",
"discounted_wealth": "V_k/B_k=V_0/B_0+sum_(j=1)^k h_j dot (S_j/B_j-S_(j-1)/B_(j-1))",
"attainable_space": "K=span{1}+L={a*1+ell:a in real numbers,ell in L}; R is not the scalar field"
}
},
"two_state_example": {
"state_order": [
"up",
"down"
],
"B_0": {
"exact": "1",
"decimal": 1.0
},
"B_1": {
"exact": "51/50",
"decimal": 1.02
},
"S_0": {
"exact": "100",
"decimal": 100.0
},
"S_1": [
{
"exact": "120",
"decimal": 120.0
},
{
"exact": "90",
"decimal": 90.0
}
],
"Q": [
{
"exact": "2/5",
"decimal": 0.4
},
{
"exact": "3/5",
"decimal": 0.6
}
],
"replication": {
"cash_account_units": {
"exact": "-750/17",
"decimal": -44.11764705882353
},
"stock_units": {
"exact": "1/2",
"decimal": 0.5
},
"initial_cash_value": {
"exact": "-750/17",
"decimal": -44.11764705882353
},
"initial_stock_value": {
"exact": "50",
"decimal": 50.0
},
"initial_cost": {
"exact": "100/17",
"decimal": 5.882352941176471
},
"terminal_cash_value": {
"exact": "-45",
"decimal": -45.0
},
"terminal_stock_values": [
{
"exact": "60",
"decimal": 60.0
},
{
"exact": "45",
"decimal": 45.0
}
],
"terminal_values": [
{
"exact": "15",
"decimal": 15.0
},
{
"exact": "0",
"decimal": 0.0
}
]
}
},
"endpoint_counterexample": {
"endpoint": "lower",
"claim_price": {
"exact": "100/51",
"decimal": 1.9607843137254901
},
"holdings_B_S_C": [
{
"exact": "5000/51",
"decimal": 98.03921568627452
},
{
"exact": "-1",
"decimal": -1.0
},
{
"exact": "1",
"decimal": 1.0
}
],
"initial_cost": {
"exact": "0",
"decimal": 0.0
},
"terminal_payoffs": [
{
"exact": "20",
"decimal": 20.0
},
{
"exact": "0",
"decimal": 0.0
},
{
"exact": "0",
"decimal": 0.0
}
]
},
"view": {
"node": "QT18-P1",
"underlying_experiment_id": "EXP-STATE-01",
"default_h": "1",
"default_V0": "0"
},
"attachments": [
{
"title": "本篇完整静态阅读",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/static/QT18-P1.html",
"kind": "html"
},
{
"title": "EXP-STATE-01 唯一冻结市场",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json",
"kind": "json",
"version": "2026-09-21-v1",
"json_pointers": [
"/units",
"/assumptions",
"/formula_contract",
"/two_state",
"/three_state_incomplete/endpoint_arbitrages"
],
"policy": "沿同一冻结文件读取当前单元指定部分，不另造树或三状态物理概率。"
},
{
"title": "同包默认精确计算",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"kind": "json",
"json_pointers": [
"/two/gain_basis",
"/two/cash_completion",
"/three/endpoints"
]
}
]
}
}
```
## Supplied entry
一期矩阵给了我们一个直观图景：所有零成本持仓的终端增益不能落进非负象限的非零部分，于是存在一组严格正的权重把它们的平均压到零。多期市场的难点是：哪些终端向量真的能由当时可知的持仓产生？即使找到了权重，为什么它会让每一期贴现价格都成为条件均值不变的过程？
这一证明单元把这两步接起来。我们先完成现金持仓的构造，再从一个有限维最近点得到严格正权重，最后用只在一个信息事件上交易的持仓恢复条件鞅性质。只需有限维线性代数、条件期望、紧性与连续函数取最小值。
## 1. 定理中的有限性与交易规则
固定 $\Omega=\{\omega_1,\ldots,\omega_m\}$，$\mathcal F=2^\Omega$，且 $P(\{\omega_i\})>0$。时点为 $0,\ldots,T$，$T<\infty$；取
$$
\mathcal F_0=\{\varnothing,\Omega\},\qquad
\mathcal F_0\subseteq\cdots\subseteq\mathcal F_T=\mathcal F.
$$
若原始样本空间更细，这里直接以终端信息的正概率原子为状态；完备性和测度唯一性都针对这些可区分状态。
现金账户 $B_k$ 为确定、严格正的有限数值；借款与存款沿同一账户增长。风险资产向量 $S_k=(S_k^1,\ldots,S_k^d)$ 取有限实值且适应滤过。允许有限、有符号的实数持仓，没有手续费、买卖价差、分红或额外借贷、卖空与流动性限制。$h_k\in\mathbb R^d$ 是在 $(k-1,k]$ 持有的风险资产单位数，必须 $\mathcal F_{k-1}$ 可测；现金账户单位数 $\beta_k$ 也一样。[^w-model]
这些条件让可实施的零成本增益构成线性空间：两套策略可以相加，也可以乘任意实数。若只允许买不能卖，这个线性结构就不能照用。
令 $\widetilde S_k=S_k/B_k$，$\widetilde V_k=V_k/B_k$。套利是初值 $V_0=0$、终值每个状态非负、至少一个状态严格正的自融资策略。由于 $P$ 全支持，这等价于 $V_T\ge0$ 且 $P(V_T>0)>0$。
## 2. 任意可预测风险持仓，都要先补上现金
对 $k=1,\ldots,T-1$，自融资要求换仓前后在同一时点计价的财富相等，没有外部注资或提款：
$$
\beta_kB_k+h_k\cdot S_k
=\beta_{k+1}B_k+h_{k+1}\cdot S_k.
$$
命题（现金补足）。给定初始财富 $V_0\in\mathbb R$ 和所有可预测风险持仓 $h_1,\ldots,h_T$，唯一的自融资现金持仓由以下递推确定：
$$
\begin{aligned}
\beta_1&=\frac{V_0-h_1\cdot S_0}{B_0},\\
\beta_{k+1}&=\frac{\beta_kB_k+h_k\cdot S_k-h_{k+1}\cdot S_k}{B_k},\qquad 1\le k\le T-1.
\end{aligned}
$$
证明。第一式是初始预算的唯一解，因为 $B_0>0$；它是 $\mathcal F_0$ 可测。若 $\beta_k$ 已构造，第二式正是上述换仓等值式对 $\beta_{k+1}$ 的唯一解。右端每项都在 $\mathcal F_k$ 可知，因此 $\beta_{k+1}$ 可用于下一期。归纳完成存在、可预测性和唯一性。[^w-cash]
现在把同一期持仓放在两个端点计价：
$$
\begin{aligned}
V_k&=\beta_kB_k+h_k\cdot S_k,\\
V_{k-1}&=\beta_kB_{k-1}+h_k\cdot S_{k-1}.
\end{aligned}
$$
各自除以对应的 $B$ 后相减，现金单位数消去，得到
$$
\widetilde V_k-\widetilde V_{k-1}
=h_k\cdot(\widetilde S_k-\widetilde S_{k-1}).
$$
逐期求和就是贴现财富恒等式：
$$
\widetilde V_k=V_0/B_0+
\sum_{j=1}^k h_j\cdot\Delta\widetilde S_j.
$$
这一步不是记号游戏。后面用到的每个“只在事件 $A$ 上持有一单位股票”的测试策略，都必须经这条递推补成完整资金策略，才真属于无套利条件所约束的对象。
## 3. 定理陈述与较直接的一边
定理（有限期第一资产定价基本定理）。在第 1 节全部模型条件下，市场无套利，当且仅当存在概率 $Q\sim P$，使每个贴现风险价格 $\widetilde S^i$ 相对于 $(\mathcal F_k)$ 是 $Q$ 鞅。有限状态中的 $Q\sim P$ 等价于每个 $q_i=Q(\{\omega_i\})$ 严格正。[^w-ftap]
先证明存在这样的 $Q$ 就没有套利。可预测性让 $h_k$ 在 $\mathcal F_{k-1}$ 下可提出，而贴现价格增量的条件均值为零。因此自融资财富满足
$$
E_Q[\widetilde V_k\mid\mathcal F_{k-1}]
=\widetilde V_{k-1}.
$$
这里所有变量都取有限多个有限值，可积性自动成立。零初值给 $E_Q\widetilde V_T=0$。若存在套利，则 $B_T>0$ 保持终值符号；某个严格正终值所在的状态也有严格正 $q_i$，使 $E_Q\widetilde V_T>0$，矛盾。
注意严格正性在这一句里真正用到了：如果 $Q$ 可以给盈利状态零权重，“均值为零”就可能看不见那个套利。
## 4. 从无套利构造严格正权重
把终端随机变量看成 $\mathbb R^m$ 中的向量，定义零初值贴现增益空间
$$
L=\left\{\sum_{k=1}^T h_k\cdot\Delta\widetilde S_k:
h_k\text{ 是可预测风险持仓}\right\}.
$$
现金补足保证这里每个向量都能由零初值自融资策略实现。它是线性子空间，因而在有限维空间中闭。无套利说 $L$ 不含非零非负向量。把这样的向量除以分量之和，可知这等价于 $L$ 与单纯形
$$
\mathcal D=\{f\in\mathbb R^m:f_i\ge0,\ \sum_i f_i=1\}
$$
不相交。下面把分离论证完整展开，而不只调用一个定理名称。[^w-separation]
先令 $G=\mathcal D-L$。它凸、非空，且不含零。它也闭：若 $x_n=f_n-\ell_n\to x$，单纯形的紧性让一个子列满足 $f_{n_j}\to f\in\mathcal D$；于是 $\ell_{n_j}=f_{n_j}-x_{n_j}\to f-x$。$L$ 闭，所以 $f-x\in L$，即 $x\in G$。
取以原点为中心、半径为 $r$ 且与 $G$ 相交的闭球 $\overline B(0,r)$。交集非空且紧，欧氏范数在其中达到最小值，记最小点为 $z$。球外所有点范数大于 $r\ge\|z\|$，故 $z$ 也是整个 $G$ 的最小范数点。由于 $0\notin G$，$z\ne0$。
对 $x\in G$ 和 $0<\lambda<1$，凸性给 $z+\lambda(x-z)\in G$。最小性于是给
$$
\begin{aligned}
\|z+\lambda(x-z)\|^2&\ge\|z\|^2,\\
2z\cdot(x-z)+\lambda\|x-z\|^2&\ge0.
\end{aligned}
$$
令 $\lambda\downarrow0$，得 $z\cdot x\ge\|z\|^2$。代入 $x=f-\ell$：
$$
z\cdot f-z\cdot\ell\ge\|z\|^2
\quad(f\in\mathcal D,\ \ell\in L).
$$
固定 $f$，把 $\ell$ 替换成任意实数倍 $a\ell$。若 $z\cdot\ell\ne0$，选适当符号、足够大的 $a$ 就使不等式失败。因此 $z\perp L$。再逐个取单纯形顶点 $f=e_i$，得到
$$
z_i\ge\|z\|^2>0\quad\text{对每个 }i.
$$
所以分离所得不是仅仅“非负”的权重。归一化
$$
q_i=\frac{z_i}{\sum_jz_j}
$$
给出全支持概率 $Q$；而 $z\perp L$ 说明 $E_Q\ell=0$ 对每个 $\ell\in L$ 成立。
## 5. 终端增益均值为零，怎样恢复每一期的条件等式
还不能直接说“因此贴现价格是鞅”。目前只知道所有零初始终端增益的 $Q$ 平均为零。我们需要把这条结论落实到每个资产、每一期、每个当时可知的事件。
固定资产 $i$、时点 $k$ 和 $A\in\mathcal F_{k-1}$，令风险持仓只在第 $k$ 期的该资产取 $\mathbf1_A$，其他期、其他资产均为零。这是可预测持仓。用第 2 节递推补足现金，初值取零，它的终端贴现增益正好为
$$
\ell=\mathbf1_A(\widetilde S_k^i-\widetilde S_{k-1}^i)\in L.
$$
因而 $E_Q[\mathbf1_A\Delta\widetilde S_k^i]=0$。这对所有 $A\in\mathcal F_{k-1}$ 都成立；再加上适应性、可积性，条件期望的积分刻画给
$$
E_Q[\widetilde S_k^i\mid\mathcal F_{k-1}]
=\widetilde S_{k-1}^i.
$$
每个资产和每一期均如此，故 $Q$ 为 EMM。至此两方向都完成。这里采用的是 Williams Lemma 3.2.6 的指标持仓思路，但已经把测试策略的现金与合法性写出。[^w-indicator]
## 6. 在冻结一期市场里看见证明中的空间
回到同一 EXP-STATE-01 二状态市场。持有一单位股票并用现金账户借款，使初值为零；贴现增益的基向量为
$$
g=\left(\frac{300}{17},-\frac{200}{17}\right),\qquad L=\operatorname{span}\{g\}.
$$
可选常数风险持仓 $h$，增益就是 $hg$。每个非零 $h$ 都让两个分量一正一负，所以 $L$ 不会进入非零非负象限。上面构造出的定价权重在此就是 $Q=\left(\frac{2}{5},\frac{3}{5}\right)$，它满足 $Q\cdot g=0$。
| 量 | 精确结果 |
|---|---|
| 风险持仓 $h$ | $1$ |
| 现金账户单位 $\beta$ | $-100$ |
| 初始财富 | $0$ |
| 终端现金金额 | $-102$ |
| 两状态财富 | $\left(18,-12\right)$ |
| 两状态贴现增益 | $\left(\frac{300}{17},-\frac{200}{17}\right)$ |
| $Q$ 下增益均值 | $0$ |
这个视图允许改变持仓和初始财富，而不改变市场价格或状态。先看现金账户单位怎样随持仓改变，再看贴现增益恒等式。改变候选 $q_u$ 也会显示：概率总和为 1 并不够，还必须使 $Q\cdot g=0$。
## 7. 迁移：未来持仓与零权重状态
问题一。仍在同一二状态市场中，设某人只在上涨结果出现时持有股票，下跌时不持有，并声称能在起点借入恰好需要的现金，使初值处处为零、终值非负。这与本定理矛盾吗？
解析。这个风险持仓是 $h_1=\mathbf1_{\{\mathrm{up}\}}$，不是平凡 $\mathcal F_0$ 可测；相应现金单位 $\beta_1=-S_0h_1/B_0$ 也需要预知结果。它确实形式上产生 $( 18,0)$ 的终值，但不是本模型允许的策略。失败的是交易信息条件，不是分离证明。现金补足只把已经可预测的风险持仓补成自融资策略，并不会替未来信息“洗白”。
问题二。三状态新增支付在价格下端 $c=\frac{100}{51}$ 时，可以找到使全部资产正确加权的 $Q=(0,\frac{9}{10},\frac{1}{10})$。为什么这不能排除套利？请核对持仓与被忽略的状态。
解析。下端套利持仓为 $(\beta,\Delta,\gamma)=\left(\frac{5000}{51},-1,1\right)$，其初始成本零、终值 $\left(20,0,0\right)$。$Q$ 恰好给唯一盈利的下跌状态零权重，所以期望仍为零。它只对 $P$ 绝对连续，不与全支持 $P$ 等价。定理必须构造每个分量严格正的权重；第 4 节逐顶点证明的正是这一点。
[^w-model]: Ruth J. Williams, [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf#page=2)，§3.1，印刷 pp.40–43 / PDF pp.2–3。章节无可靠修订日期；本文保持其有限状态、平凡初始信息和确定正现金模型。初始财富统一使用从第 1 期开始的持仓，不沿用 Lemma 3.2.3 初值行的 $\phi_0$ 排印错误。
[^w-cash]: 同章 Lemma 3.2.5 及完整证明，印刷 pp.48–49 / PDF p.6；贴现增益公式见 §3.1，印刷 pp.42–43 / PDF p.3。
[^w-ftap]: 同章 Theorem 3.2.4，印刷 pp.45–48 / PDF pp.4–6；贴现财富鞅性质见 Lemma 3.2.3，pp.44–45 / PDF p.4。
[^w-separation]: 同章 Theorem 3.6.1 及完整证明，印刷 pp.66–67 / PDF p.15。此处将其紧凸集取为单纯形，并完整给出闭性、最近点与逐分量严格正性。
[^w-indicator]: 同章 Lemma 3.2.6，印刷 pp.49–50 / PDF pp.6–7。零成本收益空间与可实施测试持仓由本文现金递推连接。
## Experiment inputs and static equivalents
```json
[
{
"id": "EXP-STATE-01",
"title": "有限市场中的状态价格、等价鞅测度与完备性：计算视图",
"anchor": "qt18-measures",
"description": "改变本视图参数后重算；正文保留默认表、完整推导和题解，公开附件提供精确输入与结果。",
"inputs": {
"shared_experiment_id": "EXP-STATE-01",
"view": {
"node": "QT18",
"underlying_experiment_id": "EXP-STATE-01",
"default_mode": "two"
},
"public_attachments": [
{
"title": "EXP-STATE-01 唯一冻结市场",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json",
"kind": "json",
"version": "2026-09-21-v1",
"json_pointers": [
"/units",
"/assumptions",
"/two_state",
"/three_state_incomplete",
"/three_state_augmented_complete"
],
"policy": "沿同一冻结文件读取当前单元指定部分，不另造树或三状态物理概率。"
},
{
"title": "同包默认精确计算",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"kind": "json",
"json_pointers": [
"/two",
"/three",
"/complete"
]
}
]
},
"outputs": {
"file": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"scope": "two/three/complete",
"json_pointers": [
"/two",
"/three",
"/complete"
]
},
"is_view_of_existing_experiment": true,
"static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/static/QT18.html"
},
{
"id": "EXP-STATE-01--na-proof",
"title": "有限期第一资产定价基本定理：无套利与等价鞅测度：计算视图",
"anchor": "qt18p1-example",
"description": "改变本视图参数后重算；正文保留默认表、完整推导和题解，公开附件提供精确输入与结果。",
"inputs": {
"shared_experiment_id": "EXP-STATE-01",
"view": {
"node": "QT18-P1",
"underlying_experiment_id": "EXP-STATE-01",
"default_h": "1",
"default_V0": "0"
},
"public_attachments": [
{
"title": "EXP-STATE-01 唯一冻结市场",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json",
"kind": "json",
"version": "2026-09-21-v1",
"json_pointers": [
"/units",
"/assumptions",
"/formula_contract",
"/two_state",
"/three_state_incomplete/endpoint_arbitrages"
],
"policy": "沿同一冻结文件读取当前单元指定部分，不另造树或三状态物理概率。"
},
{
"title": "同包默认精确计算",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"kind": "json",
"json_pointers": [
"/two/gain_basis",
"/two/cash_completion",
"/three/endpoints"
]
}
]
},
"outputs": {
"file": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"scope": "two/three/complete",
"json_pointers": [
"/two/gain_basis",
"/two/cash_completion",
"/three/endpoints"
]
},
"is_view_of_existing_experiment": true,
"static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/static/QT18-P1.html"
}
]
```
## Sources
- [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf): 有限状态模型、现金补足、指标持仓鞅判据、两条基本定理及有限维分离完整证明。数值输入属于另行冻结的教学市场，不归称讲义报价。
## Content relations
```json
[
{
"from": "zh-qt18p1",
"relation": "part_of",
"to": "quant-processes",
"reason": "主要 topic 归属"
},
{
"from": "qt18p1-example",
"relation": "illustrated_by",
"to": "EXP-STATE-01",
"reason": "同一冻结输入的当前视图；证明与算术分别呈现。"
},
{
"from": "zh-qt18p1",
"relation": "requires",
"to": "zh-qt11",
"reason": "本证明实际调用的局部能力。",
"required_competence": "使用所有信息事件上的积分身份识别条件期望。"
},
{
"from": "qt18p1-indicator",
"relation": "informs",
"to": "qt18p2-perturbation",
"reason": "正概率与指标持仓判据完成扰动后的EMM验证。"
},
{
"from": "qt18p1-model",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "Ruth J. Williams, [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf#page=2)，§3.1，印刷 pp.40–43 / PDF pp.2–3。章节无可靠修订日期；本文保持其有限状态、平凡初始信息和确定正现金模型。初始财富统一使用从第 1 期开始的持仓，不沿用 Lemma 3.2.3 初值行的 $\\phi_0$ 排印错误。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-model"
]
},
{
"from": "qt18p1-cash",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 Lemma 3.2.5 及完整证明，印刷 pp.48–49 / PDF p.6；贴现增益公式见 §3.1，印刷 pp.42–43 / PDF p.3。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-cash"
]
},
{
"from": "qt18p1-ftap",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 Theorem 3.2.4，印刷 pp.45–48 / PDF pp.4–6；贴现财富鞅性质见 Lemma 3.2.3，pp.44–45 / PDF p.4。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-ftap"
]
},
{
"from": "qt18p1-separation",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 Theorem 3.6.1 及完整证明，印刷 pp.66–67 / PDF p.15。此处将其紧凸集取为单纯形，并完整给出闭性、最近点与逐分量严格正性。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-separation"
]
},
{
"from": "qt18p1-indicator",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 Lemma 3.2.6，印刷 pp.49–50 / PDF pp.6–7。零成本收益空间与可实施测试持仓由本文现金递推连接。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-indicator"
]
}
]
```
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