What is a white noise ? What is the derivate of the Brownian motion?

Question

Could someone explain me what is a whit noise ? In my course it's written that it's the derivate of a Brownian motion, but how can it be the derivative of something that doesn't exist ?

Answer 1

Take a derivable function $f:\mathbb R\longrightarrow \mathbb R$. The definition of the derivative function is $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.\tag{$D_1$}$$

The thing is, if $f$ is derivable and $f'$ is integrable (in Lebesgue sense), then $$f(x)=f(0)+\int_0^x f'(t)dt.\tag{$D_2$}$$

In fact, you can use $(D_2)$ as a more general definiton of the derivative. I.e, $f$ is derivable if there is $g$ integrable s.t. $$f(x)=f(0)+\int_0^x g(t)dt.$$

One can prove that $g$ is unique in the sense that if $h$ is s.t. $$f(x)=f(0)+\int_0^x h(t)dt$$ then $h=g$ a.e. An other thing is if $f$ is $\mathcal C^1$ in then is derivable in $(D_2)$ sense. If $f$ is derivable in $(D_1)$ sense I would says that is also derivable in $(D_2)$ but I'm not completely sure of that (I'll check).

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Now, we can do almost the same construction with stochastic process. If $$X_t=X_0+\int_0^t f(Y_t,t)dt+\int_0^t g(Z_t,t)dB_t,$$ then we write formally the derivative of $X_t$ as $$dX_t=f(Y_t,t)dt+g(Z_t,t)dB_t.$$

Now, we have that $$B_t=\int_0^t dB_t.$$ So what we call a white noise is simply $\xi_t=dB_t$.