For every $t \in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n \geq 0$ and every $k \in \{0,1,2,...,2^n-1\},$
$$h^n_k(t) = 2^{n/2}\mathbb{1}_{[(2k)2^{-n-1},(2k+1)2^{-n-1})}(t) - 2^{n/2}\mathbb{1}_{[(2k+1)2^{-n-1},(2k+2)2^{-n-1})}(t)$$ 1. Suppose that $N_0$, ${N^n_k}_{n\geq 1, 0 \leq k\leq2^n -1 }$ are independent $N(0,1)$ random variables, Justify the existence of the (unique) Gaussian white noise $G$ on $[0,1]$, with intensity $dt$ such that $G(h_0) = N_0$ and $G(h^n_k) = N^n_k$ for every $n \geq 0$ and $0 \leq k \leq 2^n - 1$.

2.For every $t \in [0,1]$, set $B_t = G([0,t])$, verify that $$B_t = tN_0 + \sum^{\infty}_{n=0}\bigg(\sum^{2^n-1}_{k=0}g^n_k(t)N^n_k\bigg),$$ where $g^n_k(t) = \int^t_0h^n_k(s)ds$.

I am new to the brownian motion and I am self studying the stochastic process, I have known how construct the brownian motion by L^2 theory, but when I went to levy construction, I got confused about these two questions.

Could you please help me? Thank you!


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