# Testing whether a process is a Wiener process [closed]

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

Update: I was hoping to attract people who test the validity of their models (eg. that indeed a stock price follows a BS or OU model). I am more interested in the code implementation.

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

• This question is crossposted to Cross Validated where it fits better . Unfortunately it has not yet been answered. – Bob Jansen Jul 2 '17 at 17:29
• I'm voting to close this question as off-topic because see comment. – Bob Jansen Jul 2 '17 at 17:29
• I was hoping to attract people who test the validity of their models (eg. that indeed a stock price follows a BS or OU model). – OOESCoupling Jul 2 '17 at 23:37
• I am more interested in the code implementation. – OOESCoupling Jul 2 '17 at 23:38
• Here they test for OU: "Statistical Arbitrage in the U.S. Equities Market"math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb20090616.pdf – OOESCoupling Jul 2 '17 at 23:42