Intuition for Brownian motion time-inversion formula

Background: I am studying a course in Stochastic Financial Models, and have been introduced to Brownian motions (BMs) and Gaussian processes.

The BM time-inversion formula states that if $W_t$ is a BM, then $$B_t := tW_{1/t}$$ is also a BM.

I know how to prove the result, through Gaussian processes, similar to the answer here.

However, the proof is unenlightening and does not offer any intuition. In fact, my course lecturer does not offer any intuition either.

Question: Please provide some intuition why the formula “should be” true.

Example of my intuitions for another formula:

  • “Scaling” formula: if $W_t$ is a BM, then $cW_{t/c^2}$ is also a BM.
  • Intuition: picture time on the $x$ axis and the motion on the $y$ axis. BMs “don’t go that far up or down”, i.e. their standard deviation is “only” sd$(W_t)=\sqrt t$, so to compensate for the factor of $c$ stretch in the $y$ axis, we have to stretch by a factor of $c^2$ in the $x$ axis.

Answer

The idea is that we’re essentially turning the $t$-axis inside out around the point $t=1$, i.e. we’re taking $B_t$ for $1 \le t < \infty$ and compressing it into the interval $[0,1]$, and taking $B_t$ for $0 < t \le 1$ and stretching it out to the interval $[1,\infty)$. This is sort of like the reason that $\sin(1/x)$ is discontinuous at $x=0$: we’re taking all of the infinitely many oscillations between $-1$ and $1$ that $\sin(x)$ experiences on $[1,\infty)$ and compressing them down into $[0,1]$.

When we take all the variation in $B_t$ on $[1,\infty)$ and compress it into $[0,1]$, we need to rescale the size of the fluctuations to account for the fact that they’re happening on a much smaller time scale. The amount that we need to rescale by turns out to be a factor of $t$, which is probably easiest to find by making the variance correct, but makes sense: For small $t$, we need to compress $B_{1/t}$ by a lot because it came from farther along originally in the Brownian motion path and hence had more time to fluctuate.

Similarly, when we take the fluctuations in $B_t$ on $[0,1]$ and stretch them out to $[1,\infty)$ we need to rescale to make them larger. The intuition for the factor of $t$ is essentially the same: For large $t$, we need to stretch $B_{1/t}$ by a lot because it originally came from close to $0$ and hence didn’t have time to fluctuate much, so we need to amplify the fluctuations it did have by more.

Attribution
Source : Link , Question Author : Benjamin Wang , Answer Author : user6247850

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