本章导读 本章用测度变换给布朗运动加漂移，核心是 Girsanov 定理。§10.1 预备知识：10.1.1 测度（Def 10.1–10.6 测度/σ-代数/Borel σ-代数/可测空间/测度空间/概率测度）；10.1.2 绝对连续（Def 10.7 \(\nu\ll\mu\)、Def 10.8 等价测度、Def 10.9 奇异测度）；10.1.3 Radon-Nikodym 导数（Thm 10.1、Rmk 10.1 \(f=d\nu/d\mu\)）；10.1.4 概率测度变换（\(M=dQ/dP\)、\(\mathbb E_P[M]=1\)、(10.3) \(Q(E)=\mathbb E_P[\mathbf 1_E M]\)、(10.4) \(\mathbb E_Q[Y]=\mathbb E_P[YM]\)、Rmk 10.2 时变 \(M_t\)）。§10.2 给标准 BM 加漂移：两种方式（直接加 \(mdt\) vs 换测度）；\(M_t=e^{mB_t-\frac12 m^2t}\) (10.5)、\(dM_t=mM_t\,dB_t\) (10.6)、新测度 \(dQ_t=M_t\,dP\) (10.7)；Prop 10.1（\(Q_t\) 在 \(\mathcal F_s\) 上一致，可去掉下标）；Prop 10.2（在 \(Q\) 下 \(\{B_t\}\) 是漂移 \(m\) 的 BM，\(dB_t=mdt+dW_t\)）。§10.3 Girsanov 定理（(10.6) 的推广：\(dM_t=A_tM_t\,dB_t\) (10.11)、\(M_t=e^{Y_t}\)、新测度 \(P^*\)、Prop 10.3、Thm 10.2：\(W_t=B_t-\int_0^t A_s\,ds\) 在 \(P^*\) 下是标准 BM）。无图。

定义 10.1、10.2（测度、σ-代数）/ Definitions 10.1, 10.2 (Measure, σ-algebra) 测度直觉上是长度、面积、体积的推广，给每个合适的子集赋一个非负实数。定义 10.1（测度）：设 \(X\) 是集合、\(\Sigma\) 是 \(X\) 上的 \(\sigma\)-代数。函数 \(\mu:\Sigma\to\mathbb R\) 称为测度，若：1. 非负性：\(\forall E\in\Sigma\)，\(\mu(E)\ge0\)；2. 空集为零：\(\mu(\emptyset)=0\)；3. 可数可加性（\(\sigma\)-可加性）：对任意两两不交的可数集合族 \(\{E_i\}_{i=1}^\infty\subseteq\Sigma\)，\(\mu\!\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty\mu(E_k)\)。定义 10.2（\(\sigma\)-代数）：\(\Sigma\) 是 \(X\) 上的 \(\sigma\)-代数，若：\(\emptyset\in\Sigma\) 或 \(X\in\Sigma\)；\(A\in\Sigma\Rightarrow A^c=X\setminus A\in\Sigma\)；\(A_i\in\Sigma\,(i=1,2,\dots)\Rightarrow\bigcup_{i=1}^\infty A_i\in\Sigma\)。注：\(X\) 上最小的 \(\sigma\)-代数是 \(\{\emptyset,X\}\)，最大的是幂集 \(2^X\)，二者皆为平凡 \(\sigma\)-代数。A measure intuitively generalizes length, area and volume, assigning a non-negative real number to each suitable subset. Definition 10.1 (Measure): let \(X\) be a set and \(\Sigma\) a \(\sigma\)-algebra over \(X\). A function \(\mu:\Sigma\to\mathbb R\) is a measure if: 1. non-negativity: for all \(E\in\Sigma\), \(\mu(E)\ge0\); 2. null empty set: \(\mu(\emptyset)=0\); 3. countable additivity (\(\sigma\)-additivity): for any countable collection \(\{E_i\}_{i=1}^\infty\subseteq\Sigma\) of pairwise disjoint sets, \(\mu\!\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty\mu(E_k)\). Definition 10.2 (\(\sigma\)-algebra): \(\Sigma\) is a \(\sigma\)-algebra over \(X\) if: \(\emptyset\in\Sigma\) or \(X\in\Sigma\); \(A\in\Sigma\Rightarrow A^c=X\setminus A\in\Sigma\); \(A_i\in\Sigma\,(i=1,2,\dots)\Rightarrow\bigcup_{i=1}^\infty A_i\in\Sigma\). Note: the smallest \(\sigma\)-algebra over \(X\) is \(\{\emptyset,X\}\) and the biggest is the power set \(2^X\); both are trivial \(\sigma\)-algebras.

定义 10.3–10.6（Borel σ-代数、可测空间、测度空间、概率空间）/ Definitions 10.3–10.6 定义 10.3（Borel \(\sigma\)-代数）：\(\mathcal B(\mathbb R^n)\) 是 \(\mathbb R^n\) 上的 Borel \(\sigma\)-代数，若它把 \(\mathbb R^n\) 中所有开集都作为元素。定义 10.4（可测空间）：对 \((X,\Sigma)\) 称为可测空间，\(\Sigma\) 的成员称为可测集。定义 10.5（测度空间）：三元组 \((X,\Sigma,\mu)\) 称为测度空间。定义 10.6（概率测度与概率空间）：若总测度（\(X\) 上的测度）为 1，即 \(\mu(X)=1\)，则该测度是概率测度；带概率测度的测度空间是概率空间。Definition 10.3 (Borel \(\sigma\)-algebra): \(\mathcal B(\mathbb R^n)\) is the Borel \(\sigma\)-algebra over \(\mathbb R^n\) if it contains all open sets in \(\mathbb R^n\) as elements. Definition 10.4 (Measurable space): a pair \((X,\Sigma)\) is a measurable space, and the members of \(\Sigma\) are called measurable sets. Definition 10.5 (Measure space): a triple \((X,\Sigma,\mu)\) is a measure space. Definition 10.6 (Probability measure and probability space): if the total measure (the measure on \(X\)) is one, i.e. \(\mu(X)=1\), the measure is a probability measure; a measure space with a probability measure is a probability space.

定义 10.7–10.9（绝对连续、等价测度、奇异测度）/ Definitions 10.7–10.9 定义 10.7（绝对连续）：\(\mu,\nu\) 是 \((\Omega,\mathcal F)\) 上两个测度。称 \(\nu\) 关于 \(\mu\) 绝对连续，记 \(\nu\ll\mu\)，若 \(\forall E\in\mathcal F\)，\(\mu(E)=0\) 蕴含 \(\nu(E)=0\)。定义 10.8（等价测度）：\(\mu,\nu\) 等价（互相绝对连续），若 \(\nu\ll\mu\) 且 \(\mu\ll\nu\)。定义 10.9（奇异测度）：\(\mu,\nu\) 奇异，记 \(\mu\perp\nu\)，若 \(\exists E\in\mathcal F\) 使 \(\mu(E)=0\) 且 \(\nu(\Omega\setminus E)=0\)。Definition 10.7 (Absolute continuity): let \(\mu,\nu\) be two measures on \((\Omega,\mathcal F)\). We say \(\nu\) is absolutely continuous w.r.t. \(\mu\), denoted \(\nu\ll\mu\), if for all \(E\in\mathcal F\), \(\mu(E)=0\) implies \(\nu(E)=0\). Definition 10.8 (Equivalent measures): \(\mu,\nu\) are equivalent (mutually absolutely continuous) if \(\nu\ll\mu\) and \(\mu\ll\nu\). Definition 10.9 (Singular measures): \(\mu,\nu\) are singular, denoted \(\mu\perp\nu\), if there is \(E\in\mathcal F\) with \(\mu(E)=0\) and \(\nu(\Omega\setminus E)=0\).

定理 10.1（Radon-Nikodym）与注 10.1 / Theorem 10.1 (Radon-Nikodym) and Remark 10.1 定理 10.1（Radon-Nikodym 定理）：设 \(\mu,\nu\) 是可测空间 \((\Omega,\mathcal F)\) 上两个测度且 \(\nu\ll\mu\)。则存在函数（脚注 10.2：\(f\) 是 \(\Omega\) 上的随机变量）\(f:\Omega\to\mathbb R\) 使得 \(\forall E\in\mathcal F\)，\(\nu(E)=\int_{x\in E}f(x)\,d\mu(x)\) (10.1)。注 10.1：\(f\) 称为 \(\nu\) 关于 \(\mu\) 的 Radon-Nikodym 导数，记 \(f=\dfrac{d\nu}{d\mu}\)，或 \(\forall x\in\Omega\)，\(f(x)\equiv\left(\dfrac{d\nu}{d\mu}\right)(x)=\dfrac{d\nu(x)}{d\mu(x)}\) (10.2)，直觉是：任一点 \(x\) 处的 \(\nu\) 测度是 \(f(x)\) 乘以 \(x\) 处的 \(\mu\) 测度。Theorem 10.1 (Radon-Nikodym theorem): let \(\mu,\nu\) be two measures on a measurable space \((\Omega,\mathcal F)\) with \(\nu\ll\mu\). Then there exists a function (footnote 10.2: \(f\) is a random variable on \(\Omega\)) \(f:\Omega\to\mathbb R\) such that for all \(E\in\mathcal F\), \(\nu(E)=\int_{x\in E}f(x)\,d\mu(x)\) (10.1). Remark 10.1: \(f\) is called the Radon-Nikodym derivative of \(\nu\) w.r.t. \(\mu\), denoted \(f=\dfrac{d\nu}{d\mu}\), or for all \(x\in\Omega\), \(f(x)\equiv\left(\dfrac{d\nu}{d\mu}\right)(x)=\dfrac{d\nu(x)}{d\mu(x)}\) (10.2), which intuitively means the measure \(\nu\) of any point \(x\) is \(f(x)\) times the size of the measure \(\mu\) of \(x\).

概率测度变换：(10.3) 与 (10.4) / Change of probability measure: (10.3) and (10.4) 考虑概率空间 \((\Omega,\mathcal F,\mathbf P)\)。设 \(\mathbf Q\) 也是概率测度且 \(\mathbf Q\ll\mathbf P\)。由定理 10.1，\(\exists M\) 使 \(M=\dfrac{d\mathbf Q}{d\mathbf P}\)，且（因 \(\mathbf Q\) 是概率测度）\(\mathbb E_{\mathbf P}[M]=1\)。记 \(\mathbb E_{\mathbf P}[\cdot]\)、\(\mathbb E_{\mathbf Q}[\cdot]\) 为相应期望算子。则 (10.1) 蕴含 \(\forall E\in\mathcal F\)，Consider a probability space \((\Omega,\mathcal F,\mathbf P)\). Suppose \(\mathbf Q\) is also a probability measure with \(\mathbf Q\ll\mathbf P\). By Theorem 10.1, there is \(M\) with \(M=\dfrac{d\mathbf Q}{d\mathbf P}\), and (since \(\mathbf Q\) is a probability measure) \(\mathbb E_{\mathbf P}[M]=1\). Denote \(\mathbb E_{\mathbf P}[\cdot]\), \(\mathbb E_{\mathbf Q}[\cdot]\) for the respective expectation operators. Then (10.1) implies that for all \(E\in\mathcal F\),

$$\mathbf Q(E)=\int_{x\in\Omega}\underbrace{\mathbf 1\{x\in E\}}_{=\,\mathbf 1_E(x)}M(x)\,d\mathbf P(x)\;\Rightarrow\;\mathbf Q(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M].\tag{10.3}$$

而 (10.2) 蕴含对 \(\Omega\) 上任意随机变量 \(Y\)，And (10.2) implies that for any random variable \(Y\) on \(\Omega\),

$$\mathbb E_{\mathbf Q}[Y]=\int_{x\in\Omega}Y(x)\,\frac{d\mathbf Q(x)}{d\mathbf P(x)}\,d\mathbf P(x)=\int_{x\in\Omega}Y(x)M(x)\,d\mathbf P(x)\;\Rightarrow\;\mathbb E_{\mathbf Q}[Y]=\mathbb E_{\mathbf P}[YM].\tag{10.4}$$

注 10.2（时变 Radon-Nikodym 导数）/ Remark 10.2 若 \(\mathbf Q_t\) 随时间 \(t\) 演化且对任意 \(t\) 有 \(\mathbf Q_t\ll\mathbf P\)，则 (10.3) 与 (10.4) 对每个 \(t\) 成立：\(\mathbf Q_t(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\)、\(\mathbb E_{\mathbf Q_t}[Y]=\mathbb E_{\mathbf P}[YM_t]\)，其中 \(M_t=\dfrac{d\mathbf Q_t}{d\mathbf P}\)（或 \(\forall x\in\Omega\)，\(M_t(x)=\dfrac{d\mathbf Q_t(x)}{d\mathbf P(x)}\)）。If \(\mathbf Q_t\) evolves with time \(t\) and \(\mathbf Q_t\ll\mathbf P\) holds for any \(t\), then (10.3) and (10.4) hold for every \(t\): \(\mathbf Q_t(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\), \(\mathbb E_{\mathbf Q_t}[Y]=\mathbb E_{\mathbf P}[YM_t]\), where \(M_t=\dfrac{d\mathbf Q_t}{d\mathbf P}\) (or for all \(x\in\Omega\), \(M_t(x)=\dfrac{d\mathbf Q_t(x)}{d\mathbf P(x)}\)).

两种加漂移方式与 \(M_t\) / Two ways of adding drift, and \(M_t\) 设 \(\{B_t\}\) 是关于 \((\Omega,\mathcal F_t,\mathbf P)\) 的标准一维布朗运动，\(\{\mathcal F_t\}\) 由 \(\{B_t\}\) 生成。给 \(\{B_t\}\) 加漂移 \(m\) 得 \(\{W_t\}\) 有两种方式：(1) 直接、确定地加 \(mdt\)：\(dW_t=mdt+dB_t\)；(2) 把概率测度 \(\mathbf P\) 换成 \(\mathbf Q_t\)，使 \(\{B_t\}\) 在 \((\Omega,\mathcal F_t,\mathbf Q_t)\) 下是漂移为 \(m\) 的布朗运动。第一种平凡，本节聚焦第二种。先定义 Radon-Nikodym 导数Let \(\{B_t\}\) be a standard one-dimensional Brownian motion w.r.t. \((\Omega,\mathcal F_t,\mathbf P)\), with \(\{\mathcal F_t\}\) generated by \(\{B_t\}\). There are two ways to add drift \(m\) to \(\{B_t\}\) to get \(\{W_t\}\): (1) directly and deterministically add \(mdt\): \(dW_t=mdt+dB_t\); (2) change the probability measure \(\mathbf P\) to \(\mathbf Q_t\) so that \(\{B_t\}\) is a Brownian motion with drift \(m\) w.r.t. \((\Omega,\mathcal F_t,\mathbf Q_t)\). The first is trivial; this section focuses on the second. First define the Radon-Nikodym derivative

$$M_t=e^{mB_t-\frac12 m^2t},\qquad M_0=1,\quad M_t\ge0\;\forall t.\tag{10.5}$$

由例 3.2 已知 \(\{M_t\}\) 是关于 \(\mathbf P\) 的鞅。由 Ito 公式（定理 14.1）\(df(B_t)=f'(B_t)\,dB_t+\frac12 f''(B_t)\,dt\)，而鞅性意味零漂移（\(\frac12 f''(B_t)=0\)），故By Example 3.2, \(\{M_t\}\) is a martingale w.r.t. \(\mathbf P\). By Ito's formula (Theorem 14.1) \(df(B_t)=f'(B_t)\,dB_t+\frac12 f''(B_t)\,dt\), and the martingale property implies zero drift (\(\frac12 f''(B_t)=0\)), so

$$dM_t=mM_t\,dB_t.\tag{10.6}$$

在 \(\{\Omega,\mathcal F_t\}\) 上定义新概率测度 \(\mathbf Q_t\)：\(d\mathbf Q_t=M_t\,d\mathbf P\) (10.7)，或由 (10.3)，\(\forall E\in\mathcal F_t\)，\(\mathbf Q_t(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\) (10.8)。由 (10.7) 定义知 \(\forall t\)，\(\mathbf Q_t\ll\mathbf P\)。Define a new probability measure \(\mathbf Q_t\) on \(\{\Omega,\mathcal F_t\}\) by \(d\mathbf Q_t=M_t\,d\mathbf P\) (10.7), or by (10.3), for all \(E\in\mathcal F_t\), \(\mathbf Q_t(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\) (10.8). By the definition (10.7), \(\mathbf Q_t\ll\mathbf P\) for all \(t\).

命题 10.1（\(\mathbf Q_t\) 的时间一致性）/ Proposition 10.1 对 \(sFor \(s

命题 10.1 证明 / Proof of Proposition 10.1 由 (10.8)，By (10.8),

$$\mathbf Q_t(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]=\mathbb E_{\mathbf P}\!\left[\mathbb E_{\mathbf P}[\mathbf 1_E M_t\mid\mathcal F_s]\right]=\mathbb E_{\mathbf P}\!\Big[\mathbf 1_E\underbrace{\mathbb E_{\mathbf P}[M_t\mid\mathcal F_s]}_{=\,M_s}\Big]=\mathbb E_{\mathbf P}[\mathbf 1_E M_s]=\mathbf Q_s(E),$$

其中 \(\mathbf 1_E\)（\(E\in\mathcal F_s\)）可提出条件期望，\(\{M_t\}\) 鞅性给 \(\mathbb E_{\mathbf P}[M_t\mid\mathcal F_s]=M_s\)。这说明任一 \(\mathcal F_s\)-可测事件也 \(\mathcal F_t\)-可测（\(swhere \(\mathbf 1_E\) (\(E\in\mathcal F_s\)) can be pulled out of the conditional expectation, and the martingale property of \(\{M_t\}\) gives \(\mathbb E_{\mathbf P}[M_t\mid\mathcal F_s]=M_s\). This proves any \(\mathcal F_s\)-measurable event is also \(\mathcal F_t\)-measurable (\(s

去掉时间下标 / Dropping the time subscript 由命题 10.1，\(\{\mathbf Q_t\}\) 随时间推移保持过去事件的可测性，故可去掉时间下标，把 \(\mathbf Q\) 当作新概率测度。By Proposition 10.1, \(\{\mathbf Q_t\}\) maintains the measurability of past events as time passes, so we can drop the time subscript and write \(\mathbf Q\) as the new probability measure.

命题 10.2（给布朗运动加漂移）/ Proposition 10.2 (Adding drift to Brownian motion) 在概率测度 \(\mathbf Q\) 下，\(\{B_t\}\) 是漂移为 \(m\)、方差 \(\sigma^2=1\) 的布朗运动。等价地，设 \(\{W_t\}\) 是 \((\Omega,\mathcal F,\mathbf Q)\) 下的标准一维布朗运动，则 \(\{B_t\}\) 满足 \(dB_t=mdt+dW_t\)。Under the probability measure \(\mathbf Q\), \(\{B_t\}\) is a Brownian motion with drift \(m\) and variance \(\sigma^2=1\). Equivalently, let \(\{W_t\}\) be a standard one-dimensional Brownian motion w.r.t. \((\Omega,\mathcal F,\mathbf Q)\); then \(\{B_t\}\) satisfies \(dB_t=mdt+dW_t\).

命题 10.2 证明（MGF 计算）/ Proof of Proposition 10.2 只需证在 \(\mathbf Q\) 下 \(\{B_t\}\) 满足 BM 全部条件。\(\mathbf Q\)-概率 1 下 \(t\mapsto B_t\) 连续（脚注 10.4：由 \(\mathbf Q\ll\mathbf P\)——令 \(A=\{t\mapsto B_t\) 不连续\(\}\)，\(\mathbf P(A)=0\)，绝对连续给 \(\mathbf Q(A)=0\)），且 \(\mathbf Q\) 下 \(B_0=0\)。故只需证增量 i.i.d. 正态：\(s,t\ge0\) 时 \(B_{t+s}-B_s\sim\mathcal N(mt,t)\) 且独立于 \(\mathcal F_s\)。由 MGF，只需证 \(\mathbb E_{\mathbf Q}[e^{\lambda(B_{t+s}-B_s)}\mid\mathcal F_s]=e^{mt\lambda+\frac12 t\lambda^2}\)。由 (10.7) \(d\mathbf Q_{t+s\mid s}=M_{t+s\mid s}\,d\mathbf P\)，由 (10.5) \(M_{t+s\mid s}=\dfrac{e^{mB_{t+s}-\frac12 m^2(t+s)}}{e^{mB_s-\frac12 m^2s}}=\dfrac{M_{t+s}}{M_s}\) (10.9)。由 (10.4) 与 (10.9)：It suffices to show \(\{B_t\}\) satisfies all conditions of a BM under \(\mathbf Q\). With \(\mathbf Q\)-probability 1, \(t\mapsto B_t\) is continuous (footnote 10.4: from \(\mathbf Q\ll\mathbf P\) — let \(A=\{t\mapsto B_t\text{ not continuous}\}\), \(\mathbf P(A)=0\), and absolute continuity gives \(\mathbf Q(A)=0\)), and \(B_0=0\) under \(\mathbf Q\). So we only need the i.i.d. normal increment: for \(s,t\ge0\), \(B_{t+s}-B_s\sim\mathcal N(mt,t)\) independent of \(\mathcal F_s\). By the MGF it suffices to show \(\mathbb E_{\mathbf Q}[e^{\lambda(B_{t+s}-B_s)}\mid\mathcal F_s]=e^{mt\lambda+\frac12 t\lambda^2}\). By (10.7) \(d\mathbf Q_{t+s\mid s}=M_{t+s\mid s}\,d\mathbf P\), and by (10.5) \(M_{t+s\mid s}=\dfrac{e^{mB_{t+s}-\frac12 m^2(t+s)}}{e^{mB_s-\frac12 m^2s}}=\dfrac{M_{t+s}}{M_s}\) (10.9). By (10.4) and (10.9):

$$\mathbb E_{\mathbf Q}\!\left[e^{\lambda(B_{t+s}-B_s)}\mid\mathcal F_s\right]=\frac{\mathbb E_{\mathbf P}\!\left[e^{\lambda(B_{t+s}-B_s)}M_{t+s}\mid\mathcal F_s\right]}{M_s},$$

故只需证 \(\mathbb E_{\mathbf P}\!\left[e^{\lambda(B_{t+s}-B_s)}M_{t+s}\mid\mathcal F_s\right]=M_s\,e^{mt\lambda+\frac12 t\lambda^2}\) (10.10)。计算其 LHS（用 \(M_{t+s}=\frac{M_{t+s}}{M_s}M_s\)、增量独立性、\(B_{t+s}-B_s\sim\mathcal N(0,t)\) 在 \(\mathbf P\) 下的 MGF）：So it suffices to show \(\mathbb E_{\mathbf P}\!\left[e^{\lambda(B_{t+s}-B_s)}M_{t+s}\mid\mathcal F_s\right]=M_s\,e^{mt\lambda+\frac12 t\lambda^2}\) (10.10). Computing the LHS (using \(M_{t+s}=\frac{M_{t+s}}{M_s}M_s\), increment independence, and the \(\mathbf P\)-MGF of \(B_{t+s}-B_s\sim\mathcal N(0,t)\)):

$$\begin{aligned}\mathbb E_{\mathbf P}\!\left[e^{\lambda(B_{t+s}-B_s)}M_{t+s}\mid\mathcal F_s\right]&=\mathbb E_{\mathbf P}\!\left[e^{(\lambda+m)(B_{t+s}-B_s)}\mid\mathcal F_s\right]e^{-\frac12 m^2t}M_s\\&=e^{\frac{(\lambda+m)^2}2 t}\,e^{-\frac12 m^2t}M_s=e^{\frac{\lambda^2}2 t+\lambda m t}M_s,\end{aligned}$$

这正是 (10.10) 的 RHS，证毕。\(\blacksquare\)which is exactly the RHS of (10.10), completing the proof. \(\blacksquare\)

(10.6) 的推广：\(M_t\) 与新测度 \(\mathbf P^*\) / Generalizing (10.6): \(M_t\) and the new measure \(\mathbf P^*\) Girsanov 定理就是 (10.6) 中 \(M_t\) 的推广。设 \(\{B_t\}\) 是 \((\Omega,\mathcal F,\mathbf P)\) 下的标准布朗运动。类似地定义 Radon-Nikodym 导数 \(M_t\) 为 \(dM_t=A_tM_t\,dB_t\) (10.11)，或等价地 \(M_t=e^{Y_t}\)，其中由例 14.2，\(Y_t=\left[\left(\int_0^t A_s\,dB_s\right)-\dfrac12\left(\int_0^t A_s^2\,ds\right)\right]\)，\(M_0=1\)，\(M_t\ge0\)。在 \(\{\Omega,\mathcal F_t\}\) 上定义新概率测度 \(\mathbf P_t^*\)：\(d\mathbf P_t^*=M_t\,d\mathbf P\) (10.12)，或 \(\forall E\in\mathcal F_t\)，\(\mathbf P_t^*(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\) (10.13)。由定义 \(\forall t\)，\(\mathbf P_t^*\ll\mathbf P\)。Girsanov's theorem is simply a generalization of \(M_t\) in (10.6). Let \(\{B_t\}\) be a standard Brownian motion w.r.t. \((\Omega,\mathcal F,\mathbf P)\). Similarly define the Radon-Nikodym derivative \(M_t\) by \(dM_t=A_tM_t\,dB_t\) (10.11), or equivalently \(M_t=e^{Y_t}\) where, by Example 14.2, \(Y_t=\left[\left(\int_0^t A_s\,dB_s\right)-\dfrac12\left(\int_0^t A_s^2\,ds\right)\right]\), \(M_0=1\), \(M_t\ge0\). Define a new probability measure \(\mathbf P_t^*\) on \(\{\Omega,\mathcal F_t\}\) by \(d\mathbf P_t^*=M_t\,d\mathbf P\) (10.12), or for all \(E\in\mathcal F_t\), \(\mathbf P_t^*(E)=\mathbb E_{\mathbf P}[\mathbf 1_E M_t]\) (10.13). By definition \(\mathbf P_t^*\ll\mathbf P\) for all \(t\).

命题 10.3（\(\mathbf P_t^*\) 的时间一致性）/ Proposition 10.3 对 \(sFor \(s

定理 10.2（Girsanov 定理）/ Theorem 10.2 (Girsanov Theorem) 设 \(\{M_t\}\) 是由 (10.11) 定义的非负鞅，\(\mathbf P^*\) 由 (10.14) 定义。令 \(W_t=B_t-\displaystyle\int_0^t A_s\,ds\)，其中 \(\{B_t\}\) 是 \((\Omega,\mathcal F,\mathbf P)\) 下的标准布朗运动。则 \(\{W_t\}\) 是 \((\Omega,\mathcal F,\mathbf P^*)\) 下的标准布朗运动。Let \(\{M_t\}\) be the non-negative martingale defined by (10.11) and \(\mathbf P^*\) defined by (10.14). Let \(W_t=B_t-\displaystyle\int_0^t A_s\,ds\) where \(\{B_t\}\) is a standard Brownian motion w.r.t. \((\Omega,\mathcal F,\mathbf P)\). Then \(\{W_t\}\) is a standard Brownian motion w.r.t. \((\Omega,\mathcal F,\mathbf P^*)\).

定理 10.2 证明（启发式）/ Proof of Theorem 10.2 (Heuristic) 只需证在 \((\Omega,\mathcal F,\mathbf P^*)\) 下 \(dB_t=A_t\,dt+dW_t\)。设 \(\{B_t\}\) 是无穷小随机游走，则 \(dB_t=B_{t+\Delta t}-B_t=\begin{cases}\sqrt{\Delta t}&\text{prob }\frac12\\-\sqrt{\Delta t}&\text{prob }\frac12\end{cases}\) (10.15)。由 (10.11) \(dM_t=A_tM_t\,dB_t\Rightarrow M_{t+\Delta t}=M_t(1+A_t\,dB_t)\)，结合 (10.15) 得 \(M_{t+\Delta t}=\begin{cases}M_t(1+A_t\sqrt{\Delta t})&\text{prob }\frac12\\M_t(1-A_t\sqrt{\Delta t})&\text{prob }\frac12\end{cases}\)。由 (10.12) \(d\mathbf P^*_{t+\Delta t}=M_{t+\Delta t}\,d\mathbf P\) (10.16)，故It suffices to show \(dB_t=A_t\,dt+dW_t\) under \((\Omega,\mathcal F,\mathbf P^*)\). Let \(\{B_t\}\) be an infinitesimal random walk, so \(dB_t=B_{t+\Delta t}-B_t=\begin{cases}\sqrt{\Delta t}&\text{prob }\frac12\\-\sqrt{\Delta t}&\text{prob }\frac12\end{cases}\) (10.15). By (10.11) \(dM_t=A_tM_t\,dB_t\Rightarrow M_{t+\Delta t}=M_t(1+A_t\,dB_t)\), and with (10.15), \(M_{t+\Delta t}=\begin{cases}M_t(1+A_t\sqrt{\Delta t})&\text{prob }\frac12\\M_t(1-A_t\sqrt{\Delta t})&\text{prob }\frac12\end{cases}\). By (10.12) \(d\mathbf P^*_{t+\Delta t}=M_{t+\Delta t}\,d\mathbf P\) (10.16), so

$$\mathbf P^*\{B_{t+\Delta t}-B_t\}=\begin{cases}\sqrt{\Delta t}&\text{prob }\frac12\left(1+A_t\sqrt{\Delta t}\right)\\-\sqrt{\Delta t}&\text{prob }\frac12\left(1-A_t\sqrt{\Delta t}\right)\end{cases}\tag{10.17}$$

（其概率之比由 (10.16) 给出 \(\frac{M_t(1+A_t\sqrt{\Delta t})}{M_t(1-A_t\sqrt{\Delta t})}\)，且两概率之和为 1）。故由 (10.17)，\(B_{t+\Delta t}-B_t\) 在 \(\mathbf P^*\) 下的期望变化为 \(\sqrt{\Delta t}\cdot\frac12(1+A_t\sqrt{\Delta t})-\sqrt{\Delta t}\cdot\frac12(1-A_t\sqrt{\Delta t})=A_t\,\Delta t\)，启发式地说明漂移项是 \(A_t\)。\(\blacksquare\)(the ratio of the probabilities being \(\frac{M_t(1+A_t\sqrt{\Delta t})}{M_t(1-A_t\sqrt{\Delta t})}\) by (10.16), and the two probabilities summing to 1). So by (10.17), the expected change of \(B_{t+\Delta t}-B_t\) under \(\mathbf P^*\) is \(\sqrt{\Delta t}\cdot\frac12(1+A_t\sqrt{\Delta t})-\sqrt{\Delta t}\cdot\frac12(1-A_t\sqrt{\Delta t})=A_t\,\Delta t\), which heuristically shows the drift term is \(A_t\). \(\blacksquare\)