Talk:Markov process

message: Of both markovian processes and non markovian process. Please include simple clear examples. Scientifically or mathematically important examples. Examples that test the limits of the definition.   —Preceding unsigned comment added by 72.194.126.236 (talk) 08:42, 2 February 2011 (UTC)
thread_title: Needs Examples

message: I will just point out some inconsistencies and mistakes in this article:
A Markov process is, as translations from Russian state, "a process without after-effect" 
This sounds as if the word 'Markov' translates to 'without after-effect'. In fact Markov is the name of a Russian mathematician. It should be made clear to which original text the translation refers to.
From the standard definitions of drift and diffusion coefficients this means that those coefficients may depend at most on a single point (x, t) and on no history at all.
Some author seems to have mixed the concepts of 'Ito-process' and a 'Markov-process'. There is no 'drift and diffusion coefficient' of a general Markov process. The whole first paragraph is very confusing and seems to be more about Ito-processes than about Markov-processes
The Markov property is stated twice with different definitions. 
--128.130.51.57 (talk) 13:58, 26 March 2008 (UTC)
thread_title: Article needs major overhaul

message: It would be good to have an introductory sentence that introduces the subject to a layperson. For example, see . Jim 18:51, 24 March 2006 (UTC)
thread_title:  Introductory sentence 

message: For mathematicians, usually Markov process means continuous time, whereas Markov chain means discrete time. Markov processes are often used in chemistry and biology, and their properties are different than Markov chains -- would be nice to have a separate entry. Leboudec 16:24, 15 May 2006 (UTC)
replies: text: Greetings Leboudec.  Are you Jean-Yves Le Boudec, author of "Fluid Approximations of Continuous Time Markov Chains"? I've generally heard continuous time Markov processes on discrete state spaces described as Markov chains. LachlanA 02:21, 30 July 2007 (UTC)
thread_title:  Continuous Time 

message: Markov chain is only for countable set of states. I thought Markov process might contain continuous valued version.
(c.f. Mathworld ) Memming 13:15, 24 September 2006 (UTC)
thread_title:  Continuous Value 

message: Why not merge all pages but maintain them in separate sections on the Markov Chain page?  —Preceding unsigned comment added by 128.200.138.240 (talk) 18:17, 8 October 2007 (UTC)
thread_title:  Merge all, separate sections for each 

message: In the back of my mind, I tend to draw this (possibly incorrect) mental model of a Markov process as some function on a measure-preserving dynamical system, but where the underlying time evolution is not made explicit (or is not explicitly known). I think my mental model is correct (?) ... but it occurs to me: are there *any* examples of Markov processes that cannot be formulated as functions on a measure-preserving dynamical system? If so, can examples of these be given?  linas (talk) 00:15, 10 April 2008 (UTC)
replies: text: Still an open question, at least for me.  When a markov process goes stationary, the measure would be the Haar measure.  Today's version of this article now defines a markov process in terms of a filtration instead of a semigroup, but I guess filtrations indexed by natural numbers still form a semi-group, so its still a Haar measure..!?  That's what my intuition says; I've never seen this written up in pedagogical detail; any reference to this would be nice. linas (talk) 16:49, 17 July 2012 (UTC)
thread_title:  markov processes that are not dynamical systems? 

message: I found this article fairly messy and had a go at cleaning it up (I was not signed in—sorry); an expert in the area should validate that it's still correct. In particular I removed what appeared to be an extra formal definition; my understanding was that this extra definition was equivalent to the first. If the extra definition is needed for some reason, please explain the difference between the two. Ezrakilty (talk) 21:05, 19 October 2008 (UTC)
replies: text: Well, the passage of time has rendered the discussion obsolete.  The Markov chain article deals with finite probability spaces, i.e. transition matricies.  This article deals with the case where the probability space is a continuum.  While conceptually these are almost identical, there is a large shift in terminology, notation; so e.g. the use of sigma-algebras becomes obligatory instead of optional.  linas (talk) 16:42, 17 July 2012 (UTC)
thread_title:  Cleanup 

message: Is it true that given a Markovian process with probability density p[X,t], one can take the Taylor expansion in time and, in a limiting procedure, get rid of the 2nd and higher-order terms, like so,
: p[X,t+\tau] = F 
: p[X,t] + \dot{p}[X,t]\tau + \frac{1}{2}\ddot{p}[X,t]\tau^2 + \cdots = F 
: \dot{p}[X,t]\tau \approx -p[X,t] + F 
: \dot{p}[X,t] = \lim_{\tau\rightarrow0} \frac{-p[X,t] + F}{\tau} 
whereas, for a non-Markovian process, one cannot necessarily do this?  In other words, to the increasingly higher-order time terms reach information about states in the increasingly distant past?  (If so, this could be useful information to put in the Taylor series article.) Zeroparallax (talk) 07:05, 15 February 2009 (UTC)
thread_title:  Taylor expansions in time? 

message: I think that the current "formal definition" is a little weak. Given that the "state" is an uncountable set, with the current definition one may construct a processes that is not Markovian, but satisfies the condition. A better definition would be that X=\{X_t:t\geq 0\} is a Markov process if for each B \in \mathcal{B} whenever s\leq t we have that \mathbb{P}(X_t \in B | \sigma(X_k: k\leq s))=\mathbb{P}(X_t \in B | \sigma(X_s))
Does anyone agree?  —Preceding unsigned comment added by 86.176.241.179 (talk) 10:16, 31 October 2009 (UTC)
thread_title:  Formal Definition 

message: The article mentions a certain case that is a second-order Markov process, but no where does it say what would be a first-order Markov process.  If it is going to talk about "order" at all, it should define "order".
DMJ001 (talk) 04:55, 11 August 2011 (UTC)
thread_title:  Order 

message: There is a move discussion in progress on Talk:Markov chain which affects this page. Please participate on that page and not in this talk page section. Thank you.  —RMCD bot 11:30, 6 February 2017 (UTC)
thread_title: Move discussion in progress