ギルサノフの定理とハーディ空間

最近よく金融工学に使用されているギルサノフの定理を学んでいます。最初の習うべき用語は確率空間です。The probability space is an (W, F, P) triple consisting of a set of outcomes W, a sigma algebra of filtration events F, and the probability function P. もう一つ
な大事な用語は関数期間です。　Function space basically represents functions as vectors and defines a metric for distance, usually "Linf", which , for a function f, is MAX(t)|f(t)|, where |f(t)| is the norm of a vector, usually the L2. The H2 space is similar to the L2 space (it is a subset). The point of L2 in the case of functions, though, seems to refer to square-integrable functions, and the use in the Novikov and Girsanov theorems seems to be that a process Z is defined in terms of another process theta, and this process is in H2 (why H2 is chosen instead of L2 is unclear).　　ギルサノフの定理は確率分配とブラウン運動を関係する法則です。However, this brings up another strange definition, namely that of a standard Brownian motion under a measure. Brownian motion is governed by a normal distribution, which itself would be a measure, it seems.
Numerous problems remain. Among them are: why proving Z(t) = INTEGRAL(0 to t)(Z(s)*theta(s)dWs) means that Z(t) is an L2 Martingale, how Girsanov's Theorem enables pricing of assets under different probability distribution assumptions, and how to incorporate the Radon Nikodym derivative into pricing computation.
もう一つな問題は積分の書き方です。 An expectation of a function f can be written as INTEGRAL(0 to t)(f(t)dP), where P is a probability density. This is due to the fact that the integral is a Lesbegue integral, which partitions the range of the probability density and sums the width of the f value range corresponding to each probability over the range. The Riemann integral form would be INTEGRAL(0 to t)(f(t)P(t)dt).
次に金融工学によく使われている言葉です。 A martingale is a zero-drift stochastic process. In other words, E(Z(t+1)|Z(0)...Z(t)) = Z(t).
ルベーグ積分と同じように抽象的で難しい物は測度です。Measure theory is fairly broad, but, in the context of probability, the measure seems to be nothing more than the cumulative distribution function (CDF), and the Radon-Nikodym derivative, dQ/dP is the ratio of density functions. In addition, the measure Q can be written as the expectation of the R-N derivative with respect to P over the filtration, i.e. it seems that you would sum (q(x)/p(x))*p(x) over all x containing the specified set of events F (not verified).
Girsanov's Theorem supposedly indicates that, under a change of measure, the drift will change while the variance remains the same. It is used for options pricing, quantos, and the LIBOR Market Model. ギルサノフ;の定理はWがPに関する標準ブラウン運動である場合にQと言う測度が存在するしQがW~にかんする標準ブラウン運動です。W~はW-INTEGRAL(0,ｔ）（theta*ds)です。Here, theta is defined as the adapted process such that Z is defined as an integral in terms of theta, and Z is exp(INTEGRAL(0,t)(theta*dW)-INTEGRAL(0,t)*1. dQ/dP = Z. Thus, a process that is a standard Brownian Motion on P is not a standard Brownian Motion on Q, due to a change of drift, but the variances will be the same.

*1:theta*theta/2)ds