# Thread: Infinitesimal change in Wiener process proportional to the square root of time?

1. ## Infinitesimal change in Wiener process proportional to the square root of time?

I see the following statement made in a number of finance papers which indicates that the infinitesimal change in the Wiener process is proportional to the square root of (time), where the proportionality coefficient is normally distributed.

I cannot find a rigorous proof (or even a rudimentary proof) for this.

Am I missing something? Is it the case that dW is somehow just chosen to be $N\sqrt{dt}$.

TIA

2. ## Re: Infinitesimal change in Wiener process proportional to the square root of time?

Hey mkmath.

There is a property of a Weiner process in terms Wt and Wt+n which you should use to get this result [and the n > 0].

3. ## Re: Infinitesimal change in Wiener process proportional to the square root of time?

Thank you Chiro. I think my question is more fundamental than the Wiener process alone. For example, I can see how Var( $\Delta W_t$) = $\Delta t$ from the following derivation:

So assuming that $\Delta t \rightarrow dt$ for small time increments, then Var( $dW_t$) = $dt$ and $\sigma(dW_t) \rightarrow \sqrt{dt}$. At this point I am assuming the previous sentence is correct. If so, I don't understand what assumption/step is undertaken when to arrive at the statement $dW = ($some scaling factor $)*\sigma(W_t)$.

4. ## Re: Infinitesimal change in Wiener process proportional to the square root of time?

Try putting the dt factor within the variance term [since sqrt(dt)^2 = dt] and then bring it outside to show that you can have a*N(0,1) = N(0,a^2).