Stochastic Methods in Finance (1)

車天健發表於2023-02-09

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略去,詳見測度論專欄中的文章




Expectations

\(X\)\((\Omega, \mathcal{F}, P)\) 上的隨機變數,\(\mathbb{E}[X]\) 為其期望。一些期望的特殊表示如下:

  • \(X: \Omega \rightarrow \mathbb{R}\) 為簡單函式,即,\(X\) 在有限集 \(\left\{x_{1},\ldots, x_{n} \right\}\) 中取值,則:

    \[\mathbb{E}[X] := \sum\limits^{n}_{i=1} x_{i} P(X = x_{i}) \]

  • \(X \geq 0\) almost surely,則:

    \[\mathbb{E}[X] := \sup \left\{ \mathbb{E}[Y]: ~ Y \mbox{ is simple, } ~ 0 \leq Y \leq X \mbox{ almost surely. } \right\} \]

    注意,非負隨機變數的期望可能為 \(\infty\)

  • \(\mathbb{E}[X^{+}]\)\(\mathbb{E}[X^{-}]\) 其中之一是有限的,則:

    \[\mathbb{E}[X] := \mathbb{E}[X^{+}] - \mathbb{E}[X^{-}] \]

  • \(X\) 為一個向量,且 \(\mathbb{E}[|X|] < \infty\),則:

    \[\mathbb{E}\Big[\left(X_{1}, \ldots, X_{d}\right)\Big] := \Big( \mathbb{E}[X_{1}], \ldots, \mathbb{E}[X_{d}] \Big) \]




Jensen's Inequality (琴生不等式)

\(X\) 為一個隨機變數,\(g: \mathbb{R} \rightarrow \mathbb{R}\) 為一個凸函式。那麼當 \(X\) 的期望存在時:

\[\mathbb{E}[g(X)] \geq g\left(\mathbb{E}[X] \right) \]

\(g\) 為嚴格凸函式,則以上不等式可隨之寫為嚴格大於的形式(除非 \(X\) 取常數值)。


  • 注(Convex function):

    函式 \(f: X \rightarrow \mathbb{R}\) 稱作一個凸函式,如果:

    \[\forall ~ t \in [0, ~ 1]: ~ \forall ~ x_{1}, x_{2} \in X: ~ f\Big( tx_{1} + (1-t) x_{2} \Big) \leq t\cdot f(t x_{1}) + (1-t) \cdot f(x_{2}) \]




Self-Financing Condition

A self-financing strategy is defined as a consumption stream \((c_{t})_{t\geq 0}\) which follows:

\[(c_{t} - c_{t+1})\cdot P_{t} = 0 \qquad \quad \mbox{for } \forall t \geq 0 \]




Numeraire (計價單位)

  • \((\eta_t)_{t\geq 0}\) 為 previsible process.

  • \(\eta_{t} \cdot P_{t} > 0\) almost surely, i.e., \(P(\eta_t \cdot P_{t} > 0) = 1\).

  • \((\eta_{t})_{t\geq 0}\) 滿足 self-financing condition, i.e.,

    \[(\eta_{t} - \eta_{t+1}) \cdot P_{t} = 0 \qquad \quad \mbox{for } \forall t\geq 0 \]

    這實際上意味著:

    \[\eta_{t} \cdot P_{t} = \eta_{t+1} \cdot P_{t} \qquad \qquad \text{for } ~ \forall t \geq 0 \]

    注意,以上式子中兩側的 \(P_{t}\) 不能隨手約去,因為等式兩邊是兩個向量的內積運算。




Numeraire Asset

  • A numeraire asset is an asset with strictly positive price.

  • 若 asset \(i\) 為一個 numeraire asset,那麼對於 \(\forall t \geq 0\),定義 constant portfolio \(\eta\)

    \[\eta_{t}^{j} = \begin{cases} 1 \qquad \text{if } j = i\\ 0 \qquad \text{otherwise} \end{cases} \]

    為一個 numeraire portfolio。




Investment-Consumption Strategy

\[\begin{align*} c_{0} & = x - H_{1} \cdot P_{0}\\ c_{t} & = (H_{t} - H_{t+1}) \cdot P_{t} \qquad \qquad \mbox{for } t \geq 1 \end{align*} \]

其中 \(x\) 為初始財富。




Terminal Consumption Strategy

\[\begin{align*} c_{0} & = -H_{1} \cdot P_{0} = 0\\ c_{t} & = (H_{t} - H_{t+1}) \cdot P_{t} = 0 \qquad \qquad \mbox{for } 1 \leq t \leq T-1\\ c_{T} & = H_{T} \cdot P_{T} \geq 0 \\ \mbox{and} \qquad \qquad \\ P( &c_{T} > 0) > 0 \end{align*} \]

其中 \(H\) 為 previsible process,non-random \(T > 0\) 使得以上 holds almost surely。




Pure Investment Strategy

對於 \(\forall t \geq 0\),每一期持倉 \(H_{t}\),但將每一期的 consumption \(c_{t}\) 不用於消費,而是用於投資 numeraire portfolio \(\eta_{t}\)




Theorem. 區域性鞅 \(\rightarrow\) 鞅的充分條件 (local martingales to true martingales: sufficient condition)

\(X\) 為一個離散或連續的 local martingale,令過程 \((Y_{t})_{t\geq 0}\) 滿足:

\[\mbox{for } ~ \forall ~ s,t, ~ 0 \leq s \leq t: ~ |X_{s}| \leq Y_{t} \mbox{ almost surely} \]

\(\mathbb{E}[Y_{t}] \leq \infty, ~ \mbox{ for } ~ \forall ~ t \geq 0\),那麼 \(X\) 為一個 true martingale。




證明:

由於 \((X_{t})_{t\leq 0}\) 為一個 local martingale,根據定義存在一個 stopping time series (localizing sequence):\((\tau_{N})_{N\geq0}\),滿足 \(\lim \limits_{N \rightarrow \infty} \tau_{N} = \infty\),使得對於 \(\forall ~ N \geq 0\)\(\Big(X^{\tau_{N}}_{t}\Big)_{t \geq 0} = \Big(X_{t \land \tau_{N}}\Big)_{t\geq 0}\) 為 true martingale。

首先證明 \((X_{t})_{t\geq 0}\) 可積。對於任意 \(t \geq 0\),取任意 \(T \geq t\),根據條件:\(|X_{t}| \leq Y_{T}\) almost surely。又因為:\(\forall ~ T \geq 0: ~ \mathbb{E}[Y_{T}] < \infty\),那麼:

\[\mbox{for } ~ \forall ~ t \geq 0: ~ |X_{t}| \leq Y_{T} \quad \implies \quad \mathbb{E}[X_{t}] \leq \mathbb{E}[Y_{T}] < \infty \]

因此 \((X_{t})_{t\geq 0}\) integrable。

\(X_{t\land\tau_{N}}\) 視作一個下標為 \(N\) 的序列,即:

\[\Big\{ X_{t\land \tau_{N}} \Big\}_{N\geq 0} = X_{t\land \tau_{1}}, ~ X_{t\land \tau_{2}}, ~ X_{t\land \tau_{3}}, ~ \ldots \]

注意到 \(X_{t\land \tau_{N}} = X_{\min(t, \tau_{N})} \longrightarrow X_{t}\) almost surely with \(N \longrightarrow \infty\),即:

\[\mbox{for } ~ \forall ~ t \geq 0: ~ \forall ~ \varepsilon > 0: ~ P\left( \lim\limits_{N \rightarrow \infty} \left| X_{t\land \tau_{N}} - X_{t} \right| > \varepsilon \right) = 0 \]

這是因為 \(\lim \limits_{N \rightarrow \infty} \tau_{N} = \infty\)\(t \land \tau_{N} = \min(t, \tau_{N})\) 自然隨 \(N\) 增大而收斂於 \(t\)

所以對於 \(\forall ~ 0 \leq s \leq t\)

\[\begin{align*} \mathbb{E}[X_{t} ~ | ~ \mathcal{F}_{s}] & = \mathbb{E}\Big[\lim\limits_{N\rightarrow \infty}X_{t\land \tau_{N}} ~ | ~ \mathcal{F}_{s}\Big]\\ & = \lim\limits_{N \rightarrow \infty} \mathbb{E}\Big[ X_{t\land\tau_{N}} ~ | ~ \mathcal{F}_{s}\Big] \quad (\mbox{Dominated Convergence Theorem})\\ & = \lim\limits_{N \rightarrow \infty} X_{s \land \tau_{N}} \quad (\mathbf{*})\\ & = X_{s} \end{align*} \]

因此:local martingale \((X_{t})_{t\geq 0}\) 在給定的條件下也為一個 true martingale。


  • 注意:

    以上帶星號的那一步推導中,鞅 \(\Big(X_{t\land\tau_{N}}\Big)_{t\geq 0}\) 的下標依然是 \(t\),儘管現在複合為 \(t\land \tau_{N}\)。因此在這一步中我們只需將 \(t\) 替換為 \(s\) 即可。




Corollary.

假設 \(X\) 一個 離散 時間 local martingale,使對於 \(\forall ~ t \geq 0: ~ \mathbb{E}[|X_{t}|] < \infty\),那麼 \(X\) 是一個 true martingale。




證明:

\(Y_{t} = |X_{0}| + |X_{1}| + \cdots + |X_{t}|\)。Trivially:

\[Y_{t} = |X_{0}| + |X_{1}| + \cdots + |X_{t}| \geq |X_{s}| ~ \mbox{ for } ~ \forall s \in \left\{0, 1, \ldots, t \right\} \]

並且由於:\(\forall ~ t \geq 0: ~ \mathbb{E}[|X_{t}|] < \infty\),那麼:

\[\begin{align*} \mathbb{E}[Y_{t}] & = \mathbb{E}\Big[ \left|X_{0}\right| + \left|X_{1}\right| + \cdots + \left|X_{t}\right| \Big]\\ & = \sum\limits^{t}_{s=0}\mathbb{E}\big[ \left| X_{s} \right| \big] < \infty \end{align*} \]

所以 \((Y_{t})_{t\geq 0}\) 可積,並且此時 \((X_{t})_{t \leq 0}\)\((Y_{t})_{t\geq 0}\) 恰滿足上述 Sufficient Condition,因此 \((X_{t})_{t\geq 0}\) 為一個 true martingale。




Supermartingale and Submartingale (上鞅與下鞅)

上鞅(Supermartingale)

相關於 filtration \(\mathcal{\left\{ F_{t} \right\}}_{t\geq 0}\) 的一個 supermartingale(上鞅)是一個 adapted stochastic process \((U_{t})_{t\geq 0}\),滿足以下性質:

  • (Integrability)

    \[\forall ~ t \geq 0: ~ \mathbb{E}\big[\left| U_{t} \right|\big] < \infty \]

  • (Decrease in average)

    \[\forall ~ 0 \leq s \leq t: ~ \mathbb{E}\big[U_{t} ~ | ~ \mathcal{F}_{s}\big] \leq U_{s} \]




下鞅(Submartingale)

相關於 filtration \(\mathcal{\left\{ F_{t} \right\}}_{t\geq 0}\) 的一個 submartingale(下鞅)是一個 adapted stochastic process \((V_{t})_{t\geq 0}\),滿足以下性質:

  • (Integrability)

    \[\forall ~ t \geq 0: ~ \mathbb{E}\big[ | V_{t} | \big] < \infty \]

  • (Increase in average)

    \[\forall ~ 0 \leq s \leq t: ~ \mathbb{E}\big[V_{t} ~ | ~ \mathcal{F}_{s}\big] \geq V_{s} \]




鞅、上鞅、下鞅

A martingale is a stochastic process that is both a supermartingale and a submartingale.




Theorem.

假設 \(X\) 是一個連續或離散時間上的 local martingale。如果 \(X_{t} \geq 0\) 對於 \(\forall ~ t \geq 0\) 都成立,那麼 \(X\) 是一個 supermartingale(上鞅)。




證明:

\((\tau_{N})_{N\geq 0}\) 為相關於 local martingale \((X_{t})_{t\geq 0}\) 的 localizing sequence,即:

\[\forall ~ N \geq 0: ~ \Big(X^{\tau_{N}}_{t} \Big)_{t\geq 0} ~ \mbox{ is a true martingale.} \]

首先證明 \((X_{t})_{t \geq 0 }\) 可積。由 Fatou's Lemma

\[\begin{align*} \mathbb{E}\big[|X_{t}|\big] & = \mathbb{E}[X_{t}] \\ & = \mathbb{E}\Big[\lim\limits_{N \rightarrow \infty} X_{t \land \tau_{N}}\Big] \\ & = \mathbb{E}\Big[\liminf\limits_{N \rightarrow \infty} X_{t \land \tau_{N}}\Big] \\ & \leq \liminf\limits_{N \rightarrow \infty} \mathbb{E}\Big[X_{t\land \tau_{N}}\Big] \\ & = \liminf\limits_{N \rightarrow \infty} \mathbb{E}\Big[X_{t\land \tau_{N}} ~ \Big| ~ \mathcal{F}_{0} \Big] \\ & = X_{0} < \infty \end{align*} \]

在條件期望上運用 Fatou's Lemma,對於 \(\forall ~ 0 \leq s \leq t:\)

\[\begin{align*} \mathbb{E}\big[X_{t} ~ | ~ \mathcal{F}_{s}\big] & = \mathbb{E}\Big[ \lim\limits_{N \rightarrow \infty} X_{t\land \tau_{N}} ~ \Big| ~ \mathcal{F}_{s} \Big] \\ & = \mathbb{E}\Big[ \liminf\limits_{N \rightarrow \infty} X_{t\land \tau_{N}} ~ \Big| ~ \mathcal{F}_{s} \Big] \\ & \leq \liminf_{N \rightarrow \infty} \mathbb{E}\Big[ X_{t\land \tau_{N}} ~ \Big| ~ \mathcal{F}_{s} \Big] \\ & = \liminf_{N \rightarrow \infty} X_{s \land \tau_{N}} \\ & = X_{s} \end{align*} \]

因此 \((X_{t})_{t\geq 0}\) 為一個 supermartingale(上鞅)。




Corollary.

如果 \((X_{t})_{t\geq 0}\) 是一個離散時間 local martingale,且對於任意 $ t \geq 0$,有 \(X_{t} \geq 0\) almost surely,那麼 \((X_{t})_{t\geq 0}\) 是一個 true martingale。




證明:

透過上述 Theorem,我們有:

\[\mathbb{E}\big[|X_{t}|\big] = \mathbb{E}[X_{t}] \leq X_{0} < \infty \]

由於 \(X\) 是可積的,透過上一條 Corollary 可以得出 \((X_{t})_{t\geq 0}\) 是一個 martingale 的結論。




Theorem.

假設:

\[X_{t} = X_{0} + \sum\limits^{t}_{s=1} K_{s} (M_{s} - M_{s-1}) \]

其中,\(K\) 是一個 previsible process,\(M\) 是一個 local martingale,\(X_{0}\) 是一個常數。

如果對於某些非隨機的 \(T > 0\),有:\(X_{T} \geq 0\) almost surely,那麼 \((X_{t})_{0\leq t \leq T}\) 是一個 true martingale。




證明:

略。(太長了,以後有機會補上。)




隨機貼現因子(Stochastic Discount Factor / Pricing Kernel / State Price Density)

在一個沒有股息的市場中,在時刻 \(s\)\(t\) 間(\(0 \leq s < t\))的隨機貼現因子是一個 adapted positive \(\mathcal{F}_{t}-\) measurable random variable \(\rho_{s,t}\), 使得:

\[P_{s} = \mathbb{E}\big[\rho_{s,t}P_{t} ~ | ~ \mathcal{F}_{s}\big] \]




  • \(Y\) 為一個 martingale deflator(i.e. \(\forall 0 \leq s < t: ~ \mathbb{E}[Y_{t}P_{t} ~ | ~ \mathcal{F}_{s}] = Y_{s}P_{s}\)),令 \(\rho_{s,t} = \frac{Y_{t}}{Y_{s}}\),若 \(\rho_{s,t}P_{t}\) 可積,那麼 \(\rho_{s,t}\) 為時間 \(s\)\(t\) 間的 pricing kernel。

    • 證明:

      對於 positivity,由於 \(Y\) 為 martingale deflator,則 \(\forall t \geq 0: ~ Y_{t} > 0\),所以 \(\rho_{s,t} = \frac{Y_{t}}{Y_{s}} > 0\),並且:

      \[\begin{align*} \mathbb{E} \big[ \rho_{s,t} P_{t} ~ | ~ \mathcal{F}_{s} \big] & = \mathbb{E} \Big[ \frac{Y_{t}}{Y_{s}} P_{t} ~ | ~ \mathcal{F}_{s} \Big] \\ & = \frac{1}{Y_{s}} \mathbb{E} \big[ Y_{t}P_{t} ~ | ~ \mathcal{F}_{s} \big] \\ & = \frac{1}{Y_{s}} \cdot Y_{s} P_{s} \\ & = P_{s} \end{align*} \]

      因此 \(\rho_{s,t}\) 為一個 pricing kernel。

  • 相反地,對於 \(s\geq 0\),假設 \(\rho_{s, s+1}\) 為 時間 \(s\)\(s+1\) 間的 pricing kernel,令 \(Y_{t} = \rho_{0,1} \rho_{1,2} \ldots \rho_{t-1, t}\),且 \(YP\) 可積,那麼 \(Y\) 為一個 martingale deflator。

    • 證明:

      對於 \(\forall t \geq 0\),由於 pricing kernel 為正隨機變數,則 \(Y_{t} = \rho_{0,1} \rho_{1,2} \ldots \rho_{t-1, t} > 0\),並且:

      \[\begin{align*} \mathbb{E} \big[Y_{t+1}P_{t+1} ~ \big| ~ \mathcal{F}_{t} \big] & = \mathbb{E} \big[\rho_{0,1} \rho_{1,2} \ldots \rho_{t-1, t} \rho_{t, t+1} \cdot P_{t+1} ~ \big| ~ \mathcal{F}_{t} \big] \\ & = \rho_{0,1} \rho_{1,2} \ldots \rho_{t-1, t} \cdot \mathbb{E} \big[\rho_{t, t+1} \cdot P_{t+1} ~ \big| ~ \mathcal{F}_{t} \big] \qquad \text{(adaptness)}\\ & = Y_{t} \cdot P_{t} \qquad \text{(by definition)} \end{align*} \]

      因此,\((Y_{t})_{t\geq 0}\) 為一個 martingale deflator。




Proposition.

考慮存在一個 numeraire \(\eta\) 的市場,且令:\(N_{t} = \eta_{t} \cdot P_{t} \quad \forall t \geq 0\)。令 \(H\) 為一個 investment-consumption strategy,即,\(H\) 的 consumption stream 定義為:

\[\begin{align*} c_{0} & = x - H_{1} \cdot P_{0}\\ c_{t} & = (H_{t} - H_{t+1}) \cdot P_{t} \end{align*} \]

其中 \(x\) 為初始財富。令:

\[K_{t} = H_{t} + \eta_{t} \sum\limits_{s=0}^{t-1} \frac{c_{s}}{N_{s}} \]

那麼,\(K\) 為一個 pure-investment strategy from the same initial wealth \(x\)

特殊地,當且僅當 \(K\) 為一個 terminal-consumption arbitrage 時,\(H\) 為一個 arbitrage。




證明:

\[\begin{align*} (K_{t} - K_{t+1}) \cdot P_{t} & = \Big( H_{t} + \eta_{t}\sum\limits_{s=0}^{t-1}\frac{c_{s}}{N_{s}} - H_{t+1} - \eta_{t+1}\sum\limits_{s=0}^{t}\frac{c_{s}}{N_{s}} \Big) \cdot P_{t} \\ & = (H_{t} - H_{t+1}) \cdot P_{t} + \Big( \eta_{t}\sum\limits_{s=0}^{t-1}\frac{c_{s}}{N_{s}} - \eta_{t+1}\sum\limits_{s=0}^{t}\frac{c_{s}}{N_{s}} \Big) \cdot P_{t} \\ & = (H_{t} - H_{t+1}) \cdot P_{t} + \Big( \eta_{t}\sum\limits_{s=0}^{t}\frac{c_{s}}{N_{s}} - \eta_{t+1}\sum\limits_{s=0}^{t}\frac{c_{s}}{N_{s}} - \eta_{t} \frac{c_{t}}{N_{t}} \Big) \cdot P_{t} \\ & = (H_{t} - H_{t+1}) \cdot P_{t} + \Big( \big( \eta_{t} - \eta_{t+1} \big) \sum\limits_{s=0}^{t} \frac{c_{s}}{N_{s}} - \eta_{t} \frac{c_{t}}{N_{t}} \Big) \cdot P_{t} \\ & = (H_{t} - H_{t+1}) \cdot P_{t} - \eta_{t} \cdot P_{t} \frac{c_{t}}{N_{t}} + \big( \eta_{t} - \eta_{t+1} \big) \cdot P_{t} \sum\limits_{s=0}^{t}\frac{c_{s}}{N_{s}} \\ & = (H_{t} - H_{t+1}) \cdot P_{t} - \eta_{t} \cdot P_{t}\frac{c_{t}}{N_{t}} \qquad \text{(Investment-consumption strategy)} \\ & = c_{t} \cdot P_{t} - N_{t} \cdot \frac{c_{t}}{N_{t}} \qquad \text{(By definition)} \\ & = 0 \end{align*} \]

因此,對於 \(\forall t \geq 0\),有:

\[(K_{t} - K_{t+1}) \cdot P_{t} = 0 \]

由假設:\((\eta_{t})_{t\geq 0}\) 為 pure-investment strategy,則 \((K_{t})_{t\geq 0}\) 亦為 pure-investment strategy。

假設對於 non-random \(T\),有:\(c_{T} = H_{T}\cdot P_{T}\),那麼:

\[\begin{align*} K_{T} \cdot P_{T} & = \Big( H_{T} + \eta_{T}\sum\limits_{s=0}^{T-1}\frac{c_{s}}{N_{s}} \Big) \cdot P_{T} \\ & = H_{T} \cdot P_{T} + \eta_{T} \cdot P_{T} \sum\limits_{s=0}^{T-1}\frac{c_{s}}{N_{s}} \\ & = c_{T} + N_{T} \sum\limits_{s=0}^{T-1}\frac{c_{s}}{N_{s}} \\ & = N_{T} \frac{c_{T}}{N_{T}} + N_{T} \sum\limits_{s=0}^{T-1}\frac{c_{s}}{N_{s}} \\ & = N_{T} \sum\limits_{s=0}^{T}\frac{c_{s}}{N_{s}} \\ \end{align*} \]

\[\implies K_{T} \cdot P_{T} = N_{T} \sum\limits_{s=0}^{T}\frac{c_{s}}{N_{s}} \]

則:當且僅當 某些 \(c_{t} ~ (0 \leq t \leq T)\) 取值為 strictly positive 時, 等式左側 \(K_{T} \cdot P_{T}\) 為 strictly positive。




Lemma. (Bayes formula; from homework 5.)

\(\mathbb{P}\)\(\mathbb{Q}\) 為定義在 \((\Omega, ~ \mathcal{F})\) 上的 equivalent probability measures,令 Radon - Nikodym derivative: \(Z = \frac{d\mathbb{Q}}{d\mathbb{P}}\),令 \(\mathcal{G} \subset \mathcal{F}\) 為一個 \(\sigma-\)field。那麼:

\[\mathbb{E}^{\mathbb{Q}}\big[ X ~ \big| ~ \mathcal{G} \big] = \frac{\mathbb{E}^{\mathbb{P}}[ZX ~ | ~ \mathcal{G}]}{\mathbb{E}^{\mathbb{P}}[Z ~ | ~ \mathcal{G}]} \]




證明:

\(Y = \frac{\mathbb{E}^{\mathbb{P}}[ZX ~ | ~ \mathcal{G}]}{\mathbb{E}^{\mathbb{P}}[Z ~ | ~ \mathcal{G}]}\),欲證:\(\mathbb{E}^{\mathbb{Q}}\big[ X ~ \big| ~ \mathcal{G} \big] = Y\),這等價於:

對於 \(\forall G \in \mathcal{G}\)

\[\begin{align*} & \mathbb{E}^{\mathbb{Q}}\big[ X ~ \big| ~ \mathcal{G} \big] \cdot \mathbb{I}_{G} = Y \cdot \mathbb{I}_{G} \\ \iff \quad & \mathbb{E}^{\mathbb{Q}}\Big[ \mathbb{E}^{\mathbb{Q}}\big[ X ~ \big| ~ \mathcal{G} \big] \cdot \mathbb{I}_{G} \Big] = \mathbb{E}^{\mathbb{Q}} \Big[ Y \cdot \mathbb{I}_{G} \Big] \\ \iff \quad & \mathbb{E}^{\mathbb{Q}}\Big[ \mathbb{E}^{\mathbb{Q}}\big[ X \cdot \mathbb{I}_{G} ~ \big| ~ \mathcal{G} \big] \Big] = \mathbb{E}^{\mathbb{Q}} \Big[ Y \cdot \mathbb{I}_{G} \Big] \\ \iff \quad & \mathbb{E}^{\mathbb{Q}} \big[ X \cdot \mathbb{I}_{G} \big] = \mathbb{E}^{\mathbb{Q}} \Big[ Y \cdot \mathbb{I}_{G} \Big] \\ \iff \quad & \int_{G} ~ X ~ d\mathbb{Q} = \int_{G} ~ Y ~ d\mathbb{Q} \end{align*} \]

由 Radon-Nikodym derivative \(Z = \frac{d\mathbb{Q}}{d\mathbb{P}} \implies d\mathbb{Q} = Z \cdot d\mathbb{P}\)

\[\begin{align*} & \int_{G} ~ X ~ d\mathbb{Q} = \int_{G} ~ Y ~ d\mathbb{Q} \\ \iff \quad & \int_{G} ~ X Z ~ d\mathbb{P} = \int_{G} ~ YZ ~ d\mathbb{P} \\ \iff \quad & \mathbb{E}^{\mathbb{P}}\big[ XZ \cdot \mathbb{I}_{G} \big] = \mathbb{E}^{\mathbb{P}}\big[ YZ \cdot \mathbb{I}_{G} \big] \end{align*} \]

因此,目標等價於證明:對於 \(\forall G \in \mathcal{G}\),有:

\[\mathbb{E}^{\mathbb{P}}\big[ XZ \cdot \mathbb{I}_{G} \big] = \mathbb{E}^{\mathbb{P}}\big[ YZ \cdot \mathbb{I}_{G} \big] \]

注意到 \(Y = \mathbb{E}^{\mathbb{Q}}\big[ X ~ \big| ~ \mathcal{G} \big]\)\(\mathcal{G}-\)measurable,那麼RHS:

\[\begin{align*} \mathbb{E}^{\mathbb{P}}\big[ YZ \cdot \mathbb{I}_{G} \big] & = \mathbb{E}^{\mathbb{P}} \Big[ \mathbb{E}^{\mathbb{P}}\big[ YZ \cdot \mathbb{I}_{G} ~ \big| ~ \mathcal{G} \big] \Big] \qquad \text{(Tower property)} \\ & = \mathbb{E}^{\mathbb{P}} \Big[ \mathbb{I}_{G}Y \cdot \mathbb{E}^{\mathbb{P}}\big[ Z ~ \big| ~ \mathcal{G} \big] \Big] \qquad \text{($\mathbb{I}_{G}Y$ is $\mathcal{G}-$measurable)} \\ & = \mathbb{E}^{\mathbb{P}} \Big[ \mathbb{I}_{G} \cdot \frac{\mathbb{E}^{\mathbb{P}}[ZX ~ | ~ \mathcal{G}]}{\mathbb{E}^{\mathbb{P}}[Z ~ | ~ \mathcal{G}]} \cdot \mathbb{E}^{\mathbb{P}}\big[ Z ~ \big| ~ \mathcal{G} \big] \Big] \\ & = \mathbb{E}^{\mathbb{P}} \Big[ \mathbb{I}_{G} \cdot \mathbb{E}^{\mathbb{P}} \big[ZX ~ \big| ~ \mathcal{G} \big] \Big] \\ & = \mathbb{E}^{\mathbb{P}} \Big[ \mathbb{E}^{\mathbb{P}} \big[ZX \cdot \mathbb{I}_{G} ~ \big| ~ \mathcal{G} \big] \Big] \qquad \text{($\mathbb{I}_{G}$ is $\mathcal{G}-$measurable)} \\ & = \mathbb{E}^{\mathbb{P}} \big[ ZX \cdot \mathbb{I}_{G} \big] \qquad \text{(Tower property)} \end{align*} \]

證畢。

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