# Testing whether a process is a Wiener process

Ideally I would like links to code implementations (eg. Matlab ) or book references, but I would appreciate suggestions on various methods.

We start with sampled process $X_{t}$.

1. A straightforward way is through repeated normality tests:

a. Test for normality of $X_{t}$ using KS test

b. Independence of increments: we test whether the product $X_{T/2}(X_{T}-X_{T/2})/\sigma^{2}$ is close to a standard normal, where $\sigma^{2}=(T/2)^{2}$ .

c. Joint normal for increments $Y_{n}=(X_{t_2}-X_{t_1},...,X_{t_n}-X_{t_n-1})$: we divide the possible values of $Y_{n}$ into $m$ cells and denote $O_{j}:=$# samples that fall into cell $j$. Then the statistic $\sum \frac{(O_{j}-E[O_{j}])^{2}}{E[O_{j}]}$ should be approximately a $\chi^{2}-$distribution.

2. Maybe testing for quadratic variation=$t$ and martingale property (Levy characterization)?

3. Some spectral characterization for WP?

4. The other link is testing for fractional WP, which is more involved.

5. Turning it into a test for White noise since it is the "derivative" of WP. So maybe taking finite difference for WP and doing white noises tests for it: $$(X_{t+\Delta t}-X_{t})/\Delta t.$$

• I would go with something similar to number 5. If you have all of the samples in question from $X_t$, do an autocorrelation and take the Fourier transform and see if the power spectral density is inversely proportional to $\omega^2$. More information at the wiki en.wikipedia.org/wiki/Brownian_noise Jul 14 '17 at 18:50

That's not a good question to ask (you are ok and I'll vote up your question in a sec). BM induces a measure on functions, called the wiener measure. Since this is a measure on functions, it is infinite dimensional. The thing with measures on infinite dimensional spaces is that they tend to be mutually singular: events that happen w.p. 1 wrt one measure may have zero probability wrt the other. For example, take BM $B_t$ and BM with some tiny constant added $C_t= B_t + \epsilon$. Then $P(B_0 = 0)=1$ but $P(C_0=0)=0$. This situation is fairly generic.