ما معنى Change of measure؟

Example 1.16 (Shifting a Gaussian, and seeing a far tail)

Let $X \sim \mathcal N(0, 1)$ under $\P$ and $Z = \exp(cX - c^2/2)$. Then $\E^{\mathbb Q}[e^{\iu uX}] = \E[e^{(c + \iu u)X - c^2/2}] = e^{\iu uc - u^2/2}$: under $\mathbb Q$, $X \sim \mathcal N(c, 1)$. The change of measure has moved the mean without touching the shape. Run it backwards to estimate $p = \P(X > 4) = 3.17 \times 10^{-5}$: sample $Y \sim \mathcal N(4, 1)$ and average $e^{-4Y + 8}\mathbf 1_{\{Y > 4\}}$, the indicator reweighted by $d\P/d\mathbb Q$. With 100 000 draws, plain sampling sees three exceedances and has a standard error of $1.7 \times 10^{-5}$, half the answer; the reweighted estimate has a standard error of $2.1 \times 10^{-7}$, eighty times smaller. Chapter 26 turns this into importance sampling, and chapter 5 does the same computation for whole Brownian paths.