伊藤等長と二次変動

最近確率的な積分を学んでいます。I have had difficulty in deriving the expression sometimes given for the quadratic variation of the Ito integral. It seems to be based on Ito's isometry, which I have yet to see the derivation of.
一つな前提はB（ｔ）と言うブラウン運動は微分ができない関数です。無作為な値を出力するからです。
Ito's isometry expresses the expectation of the square of a stochastic integral for a function in terms of the standard integral of the square of that function.
E*1

It is unclear how such a formula is derived, but it appears to arise from the definition of covariance for stochastic functions.
Another formula, an unproven identity, relates quadratic variance of an Ito integral to the Ito integral of a function with respect to the quadratic variance of Brownian motion.
[(INT(0,T)(x(t)dB(t)))] = (INT(0,T)(x(t)d[B(t)]))
Here, [B(t)] is the quadratic variation of B.
The use of stochastic integrals seems to be to integrate stochastic differential equations, such as those used in geometric Brownian motion. Ito's lemma does not involve stochastic integrals, but the SDE's it translates between are solved via them. A more difficult concept is quadratic variation. It seems to be a value [M] such that M^2(t)-[M](t) is a Martingale. In addition, the variance of M is E([M]), which makes little sense, at first. However, if [M](0) is defined to be 0 (as it is in many places), then E(M^2(t)-[M](t)) = E(M^2(t))-[M](0) . Since the variance of M is E(M^2(t))-(E(M(t))^2), var(M) = E(M^2(t))-M^2(0), as M is a martingale. As M^2(t)-[M](t) is a Martingale by definition of [M], E(M^2(t))-E([M](t)) = E(M^2(0))-[M](0). As [M](0)=0, E(M^2(t))-E([M](t)) = E(M^2(0)). As E(M(t)) = M(0), E(M^2(t))-E(M^2(0)) = var(M) = E([M]), so the variance of the Martingale process is equal to the expectation of the quadratic variation of the process. This can be extended to demonstrate the variance of the Ito integral. var(I(t)) = E([I](t)), which is defined by another theorem.
派生証券の値を計算する事に役に立ちそう。

credit
https://almostsure.wordpress.com/2010/03/29/quadratic-variations-and-the-ito-isometry/, Wikipedia

*1:INT(0,T)(x(t)dB(t)))^2) = E(INT(0,T)(x^2(t)dt