ギルサノフの定理の多形式

最近ギルサノフの定理の由来と使い方を考え続けている。 One form involves integrals, while the other involves the covariance notation.
Wikipedia states that Y~ = Y - [Y,X] is a Q local martingale, given that Y is a local martingale under P. Also W~ = W - [W,X] is standard Brownian Motion under Q, where W is standard Brownian Motion under P.
シカゴ大学の統計学講座の資料によると
W~ = W - INTEGRAL(s=0 to t)(theta*ds)
http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf
以下はこの論本に基づく。http://www.qmss.jp/prob/stochasticproc/35-girsanov.pdf
The point of Girsanov's Lemma is to define, for a given market process V and measure P, under which V is not standard Brownian motion, a new measure Q exists such that, under Q, V is a standard Brownian Motion. dQ/dP is known as the Radon-Nikodym derivative and itself is equivalent to a Martingale stochastic process (確立過程).
(dQ/dP) = exp(-INTEGRAL(0 to s)(u(t))dW(t) - INTEGRAL(0 to s)(u(t)**2)dt
Q = INTEGRAL(S)(VdP), where S is a subset of the time space, it seems. The numeraire, or 基準財、is also related, as, if the numeraire for a Martingale process f/g g is the riskless asset, then the equivalent Martingale measure is the risk-neutral measure.
論文に分からなかった事は対数正規分布の過程がマーチンゲールと言う事です。
具体的にB(t)がブラウン運動であればexp(B(t)-t/2)がマーチンゲールです。
Note that the Girsanov theorem is a special case of the Cameron-Martin Theorem, which also defines a Radon-Nikodym derivative as an exponential.
One difficulty with Girsanov's lemma is that it is defined only for transitioning between equivalent probability measures. In addition, it seems to provide no exact formula for translating from a P expectation to a Q expectation. In fact, though taking an expectation wrt P is integrating wrt to P of f, as P is the cumulative distribution function and dP is the probability density, or . Thus E(wrt Q)f = INTEGRAL(fdP(dQ/dP)), so the Radon-Nikodym derivative enables measure change.
ラドンニコディム微分を見つければ測度変換が出来る。
Application domains are swaptions (risk neutral with respect to the annuit numeraire) and the LIBOR Market Model (rolling forward risk neutral world). Note that, in the case of a swaption, the change of measure makes the swap rate, not the swaption price, be a Martingale. It seems that the numeraire determines the measure, so it must be related to the Radon-Nikodym derivative. Change of measure seems to be due to the change in the timing of the uncertain future cash flows. In addition, the measure changes when converting from risk-neutral to historical probabilities.