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

## closed as off-topic by Bob Jansen♦Jul 2 '17 at 17:29

• This question does not appear to be about quantitative finance within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• 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