Testing whether a process is a Wiener process

Question

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: $$(X_{t+\Delta t}-X_{t})/\Delta t.$$

Answer 1

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.

What this means to you is that unless your process really is exactly BM (highly unlikely), as you take more and more data points your measure will converge to something that is mutually singular to BM and your hypothesis test will give a "no". This may happen even if your process is pretty "similar" to BM.

That's why this is not a good question to ask. Measures over function spaces are tough to work with.