Construction of Brownian Motion using Haar wavelets

Question

I want to construct brownian motion using Haar Wavelets where $\varphi_n, n \in \mathbb{N}$ is an orthonormal basis of $L^2[0,1]$. I take the inner product: $$\langle f,g \rangle = \int_0^1 f(t)g(t)dt$$ Now I take $$\Phi_n(t) = \int_0^t \varphi_n(s) ds$$ This is obviously well defined since $\varphi_n \in L^2[0,1] \subset L[0,1]$.

Now I take a series of i.i.d. random normal variables $Z_n$ to create $$W_N^\varphi(t)= \sum_{n=1}^N Z_n \Phi_n(t)$$

I have shown that this is a martingale (w.r.t $\mathcal{F}_N$) for all $t \in [0,1]$. Furthermore I have shown (by the martingale convergence theorem) that $$W^\varphi(t)=\lim_{N \rightarrow \infty} W_N^\varphi(t) = \sum_{n=1}^\infty Z_n \Phi_n(t)$$ is well defined and Gaussian.

I now want to show that this has independent increments and that for $0\leq s\leq t \leq 1$: $$W^\varphi(t)-W^\varphi(s) \sim N(0,t-s)$$ Since $W^\varphi(t)$ is Gaussian, it is sufficient to show that the increments are uncorrelated, but I also don't know how to do that.

Help would be greatly appreciated!

Answer 1

Using the independence of $Z_n \sim N(0,1)$ we find

$$\begin{align*} \mathbb{E}\big( (W_N(t)-W_M(t))^2 \big) &= \mathbb{E} \left( \sum_{n,m=M+1}^N Z_n Z_m \Phi_m(t) \Phi_n(t) \right) \\ &= \sum_{n,m=M+1}^n \underbrace{\mathbb{E}(Z_n Z_m)}_{\delta_{n,m}} \Phi_m(t) \Phi_n(t) \\ &= \sum_{n=M+1}^N \Phi_n(t)^2. \end{align*}$$

By Parseval's identity, the right-hand side converges to $0$ as $M,N \to \infty$, and therefore we find that $(W_N(t))_{N \geq 1}$ is a Cauchy sequence in $L^2(\mathbb{P})$, hence $W(t) = L^2-\lim_{N \to \infty} W_N(t)$. Using the above calculation for $M=0$, we get

$$\mathbb{E}(W(t)^2) = \lim_{N \to \infty} \mathbb{E}(W_N(t)^2) = \sum_{n=1}^{\infty} \Phi_n(t)^2 = t.$$

Similarly, we find for any $s

$$\begin{align*} \mathbb{E}((W(t)-W(s))(W(v)-W(u))) &= \lim_{N \to \infty} \mathbb{E}((W_N(t)-W_N(s))(W_N(v)-W_N(u))) \\ &= \lim_{N \to \infty} \mathbb{E} \left( \sum_{m,n=1}^N (\Phi_n(t)-\Phi_n(s)) (\Phi_m(v)-\Phi_m(u)) Z_m Z_n \right) \\ &= \sum_{n=1}^{\infty} (\Phi_n(t)-\Phi_n(s)) (\Phi_n(v)-\Phi_n(u)) \\ &= \int_0^1 1_{[s,t)}(x) 1_{[u,v)}(x) \, dx. \end{align*}$$

Obviously, the right-hand side equals $0$ if $[s,t) \cap [u,v) = \emptyset$; this shows that $W(t)-W(s)$ and $W(v)-W(u)$ are uncorrelated; hence, independent.

Reference: Schilling & Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Section 3.1.