Talk:Ornstein-Uhlenbeck process

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The definition of the Ornstein-Uhlenbeck is not correct. The Ornstein-Uhlenbeck process does not have to be mean reverting, it can have a general drift process, i.e.

dr_t = \mu\,dt + \sigma\, dW_t,\,

However, it is possible to define mean-reverting Ornstein-Uhlenbeck processes like

dr_t = -\theta (r_t-\mu)\,dt + \sigma\, dW_t,\,

Usually an Ornstein-Uhlenbeck process refers to processes, where the process itself does not appear in the stochastic part of the SDE which describes the dynamics of the process, i.e. σ does not depend on r.


I disagree. The simplest definition of an OU process is given by dr_t = -\theta r_t \,dt + \sigma\, dW_t,\, which is mean reverting to the zero, (did you have a typo in your definition of the OU process?). The definition in the article is just a more general case of the OU process than we are normally used to seeing. If you want, just set μ = 0.
see also http://pauillac.inria.fr/algo/csolve/ou.pdf. If you think this is incorrect, could please you provide a better reference?


Covariance function for this process looks strange and quite different from earlier versions of this article, released last summer in particular.

Indeed, when |s-t| -> oo, cov(rs, rt) should be 0, as autocorrelation effect decreases with time interval.

The article fails to mention what dWt is!

Indeed. I added that Wt refers to the Wiener process. -- Jitse Niesen (talk) 05:49, 26 June 2006 (UTC)

[edit] text below the graph

The text below the graph should read \theta\ = 1, \mu\ =1.2 , \sigma\ = 0.3 to be consistent with the legend on the actual graph itself. I made this correction on Aug. 30, 2006 (ko4seki)

The text below the graph gives values for the parameter a but fails to mention what it is. Tim (talk) 07:43, 7 January 2008 (UTC)