Levy construction of Brownian motion by Haar function and Schauder function
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For every $t in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n geq 0$ and every $k in 0,1,2,...,2^n-1,$
$$h^n_k(t) = 2^n/2mathbb1_[(2k)2^-n-1,(2k+1)2^-n-1)(t) - 2^n/2mathbb1_[(2k+1)2^-n-1,(2k+2)2^-n-1)(t)$$
1. Suppose that $N_0$, $N^n_k_ngeq 1, 0 leq kleq2^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
$$B_t = tN_0 + sum^infty_n=0bigg(sum^2^n-1_k=0g^n_k(t)N^n_kbigg),$$
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!
probability-theory stochastic-processes brownian-motion levy-processes
asked Sep 11 at 2:40
Rachel Buiter
63
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up vote
1
down vote
favorite
For every $t in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n geq 0$ and every $k in 0,1,2,...,2^n-1,$
$$h^n_k(t) = 2^n/2mathbb1_[(2k)2^-n-1,(2k+1)2^-n-1)(t) - 2^n/2mathbb1_[(2k+1)2^-n-1,(2k+2)2^-n-1)(t)$$
1. Suppose that $N_0$, $N^n_k_ngeq 1, 0 leq kleq2^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
$$B_t = tN_0 + sum^infty_n=0bigg(sum^2^n-1_k=0g^n_k(t)N^n_kbigg),$$
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!
probability-theory stochastic-processes brownian-motion levy-processes
asked Sep 11 at 2:40
Rachel Buiter
63
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
For every $t in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n geq 0$ and every $k in 0,1,2,...,2^n-1,$
$$h^n_k(t) = 2^n/2mathbb1_[(2k)2^-n-1,(2k+1)2^-n-1)(t) - 2^n/2mathbb1_[(2k+1)2^-n-1,(2k+2)2^-n-1)(t)$$
1. Suppose that $N_0$, $N^n_k_ngeq 1, 0 leq kleq2^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
$$B_t = tN_0 + sum^infty_n=0bigg(sum^2^n-1_k=0g^n_k(t)N^n_kbigg),$$
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!
probability-theory stochastic-processes brownian-motion levy-processes
asked Sep 11 at 2:40
Rachel Buiter
63
For every $t in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n geq 0$ and every $k in 0,1,2,...,2^n-1,$
$$h^n_k(t) = 2^n/2mathbb1_[(2k)2^-n-1,(2k+1)2^-n-1)(t) - 2^n/2mathbb1_[(2k+1)2^-n-1,(2k+2)2^-n-1)(t)$$
1. Suppose that $N_0$, $N^n_k_ngeq 1, 0 leq kleq2^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
$$B_t = tN_0 + sum^infty_n=0bigg(sum^2^n-1_k=0g^n_k(t)N^n_kbigg),$$
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!
probability-theory stochastic-processes brownian-motion levy-processes
probability-theory stochastic-processes brownian-motion levy-processes
asked Sep 11 at 2:40
Rachel Buiter
63
asked Sep 11 at 2:40
Rachel Buiter
63
asked Sep 11 at 2:40
Rachel Buiter
63
asked Sep 11 at 2:40
Rachel Buiter
63
asked Sep 11 at 2:40
Rachel Buiter
63
63
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add a comment |
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