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Sampling Fixed-Time Iterated Brownian Motion Values

Article Quant Q&A · Author: Rishabh Kumar
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Summary

The document proposes a recursive way to simulate the value of an iterated Brownian motion at a fixed time. Start with a normally distributed Brownian value whose variance is the time horizon; at each subsequent iteration, draw another centered normal value with variance equal to the absolute value of the previous draw. This gives a sequence of nested random values associated with the process at that time.

The answer provides a conceptual sampling recipe rather than code or numerical validation. It addresses a single time point, so it does not explain how to generate a consistent path over many time points, nor does it establish that the shorthand notation in the response fully specifies the iterated process. Anyone implementing a path simulation would need to account for dependence between time points and verify the intended definition of the nested Brownian motions against a formal source.

Key ideas

  • At a fixed time, a Brownian motion value has a centered normal distribution with variance equal to that time.
  • The proposed recursion uses the absolute value of the preceding draw as the next draw's variance.
  • Repeating this sampling step gives a proposed method for generating an iterated process value.
  • The recipe describes fixed-time values and does not specify a dependent path simulation.

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# Simulating Iterated Brownian Motions


# Simulating Iterated Brownian Motions












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:

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

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/71257

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.