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I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) and stumbled across something similar; iterated Brownian motions. Iterated Brownian motions have been defined as follows: enter image description here

I was wondering how to simulate, say, $W_2(t)$, or in general $W_n(t)$ on a computer?

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If I understand this correctly, we could simulate this process as follows. Let $N(0,t)$ denote the the Normal distribution with variance $t$.

Given some fixed level $t$, simulate

  • $B_1(t)\sim N(0,t)$, i.e. the first iterate is a typical Brownian motion for time horizon $t$.
  • Then, $B_2(t)\sim N(0,|B_1(t)|)$ is a Brownian motion (normally distributed RV) with variance $|B_1(t)|$.
  • Repeat for all $i\leq n$: $B_i(t)\sim N(0,|B_{i-1}(t)|)$
  • Set $W_n(t)=B_n(\ldots)$

HTH?

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  • $\begingroup$ Ohh, it sounds good to me. Thanks for the reply :D. I think it was straightforward, now that I see it. $\endgroup$ Commented Jun 15, 2022 at 15:53

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