測度變換(Change of measure)

• 測度(measure)為度量(metric)的推廣。
• 比如說在一維空間的測度為距離，二維空間的測度為面積，三維空間的測度為體積等。
• 一般生活中常使用單位換算，可視為測度變換的特例。
  • E.g. 公分 $\Leftrightarrow$ 公尺，美金$\Leftrightarrow$日元，攝氏$\Leftrightarrow$華氏。
• 因為物質的本質不變，所以即使用不同單位量測，本質仍然不變，只有量測值會變動。
• 上述例子均為實數的變換，這邊要討論的測度變換為random variable的變換。

What is a change of the underlying measure?

• The main idea of the change of measure technique consists of introducing a new probability measure via a so-called density function which is in general not a probability.

絕對連續(測度) (absolutely continuous (measure))

• $P$, $Q$ are two probability measures on the same sigma-field $F$。
• If there exists a non-negative function $g_1$ s.t.
  • $Q(A) = \int_A g_1(\omega)dP(\omega), \forall A \in F$.
  • We say that $g_1$ is the density of $Q$　w.r.t. $P$.
  • $Q$ is absolutely continous w.r.t. $P$ (記為 $Q << P$).
  • 絕對連續即機率測度$P$可經由一正值函數$g_1$轉換成機率測度$Q$。
  • $g_1$必須為正值是因為機率測度值必須大於等於0。

等價測度 (Equivalent probability measure)

• If $P$ is absolutely continuous w.r.t. $Q$ ($P << Q$) and
• If $Q$ is absolutely continuous w.r.t. $P$ ($Q << P$).
• Then $P$ and $Q$ are equivalent probability measures.
  • i.e. $P(A) > 0 \Leftrightarrow Q(A) > 0$.
  • 即原本$Q$測度為0的事件集合，使用$P$測度仍然為$0$，而$Q$測度不為0的事件集合，使用$P$測度不一定為0。

Example: Normal distribution

• Considering $X \sim N(\mu, \sigma^2)$
  • The pdf $f_{\mu,\sigma^2}(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{(x-\mu)^2}{2\sigma^2}}$.
  • The cdf $F_{\mu, \sigma^2}(x) = \int_{-\infty}^{x} f_{\mu, \sigma^2}(y) dy$.
• Consider two pairs parameters $(\mu_1, \sigma_1^2)$, $(\mu_2,\sigma_2^2)$.
  • Define $g_1=\frac{f_{\mu_1, \sigma_1^2}(x)}{f_{\mu_2, \sigma_2^2}(x)} > 0$, $g_2=\frac{f_{\mu_2, \sigma_2^2}(x)}{f_{\mu_1, \sigma_1^2}(x)} > 0$.
  • Then $F_1(x) = \int_{-\infty}^{x}g_1(y)f_{\mu_2, \sigma_2^2}(y)dy$.
  • $F_2(x) = \int_{-\infty}^{x}g_2(y)f_{\mu_1, \sigma_1^2}(y)dy$.
• 可知 $g_1$, $g_2$不是pdf，但是為正值的函數。

Example: Exponential function

• $P \sim exp(\mu)$，$Q \sim exp(\lambda)$.
• $\therefore$ $P$，$Q$ are equivalent measures。
• $\begin{array}{rcl} e_Q(X) & = & \int_{\Omega}X dQ \\ & = & \int_{\Omega}x\lambda e^{-\lambda x} \\ & = & \int_{\Omega} \frac{\lambda e^{-\lambda x}}{\mu e^{- \mu x}}e^{-\mu x} dx \\ & = & E_p(ZX) \end{array}$
• $Z = \frac{\lambda e^{-\lambda x}}{\mu e^{- \mu x}}$ 稱為Raydon-Nikodym derivative.

Randon-Nikodym derivative

• If $P$ and $Q$ are equivalent measures, and $X_t$ is an $F_t$-adapted process, then
  • $E_Q(X_t) = E_p \left( \frac{dQ}{dP}X_t \right)$.
  • $E_Q(X_t \vert F_t) = L_s^{-1} E_p \left( L_t X_t \vert F_s \right)$.
    • $L_s = E_p \left( \frac{dQ}{dP}X_t \vert F_s \right)$.
• $L_t$ is the Radon-Nikodym derivative of $Q$ with respect to $P$.