What is the polarization identity?

Question

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and meaning. Could someone give me the mathematical definition?

Answer 1

The polarization identity holds for any scalar product $\langle \cdot,\cdot \rangle$:

$$\langle x,y \rangle = \frac{1}{4} \big( \langle x+y,x+y \rangle - \langle x-y,x-y \rangle \big).$$

In $\mathbb{R}$ this equality boils down to

$$x \cdot y = \frac{1}{4} \big( (x+y)^2-(x-y)^2 \big). \tag{1}$$

One important application in stochastic calculus is a generalization of Itô's isometry: In fact, using $(1)$, it follows easily that

$$\mathbb{E} \left( \left[ \int_0^t f(s) \, dB_s \right]^2 \right) = \mathbb{E} \int_0^t f(s)^2 \, ds$$

implies

$$\mathbb{E} \left( \int_0^t f(s) \, dB_s \cdot \int_0^t g(s) \, dB_s \right) = \mathbb{E} \int_0^t f(s) \cdot g(s) \, ds.$$

Answer 2

This is the way to get the symmetric bilinear application associated to a quadratic form.

There are several of them, including \begin{align} 4\phi(x,y) &= Q(x+y)-Q(x-y)\\ 2\phi(x,y) &= Q(x+y) - Q(x) - Q(y)\\ 2\phi(x,y) &= Q(x) + Q(y) - Q(x-y) \end{align}