Is Markov process a Brownian process?

  • #1

Main Question or Discussion Point

Hi all,

I know that Brownian process can be shown as Markov process but is the converse possible? I mean can we show that a markov process is a brownian process?

Thanks in advance.
 

Answers and Replies

  • #2
mathman
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There are discrete processes with the Markov property. Brownian motion is continuous.
 
  • #3
There are discrete processes with the Markov property. Brownian motion is continuous.
Sorry, I didn't understand.
The book I am reading shows Brownian as Markov.
 
  • #4
lavinia
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Hi all,

I know that Brownian process can be shown as Markov process but is the converse possible? I mean can we show that a markov process is a brownian process?

Thanks in advance.
Not sure what you mean by a Brownian process but if you mean a Weiner process then there are many Markov processes that are not Weiner processes. For instance,in finance, geometric Brownian motions are commonly use to model securities prices.
 
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  • #5
Borek
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Brownian motion is continuous.
I believe in the scale large enough Brownian motions are indistinguishable from (continuous) diffusion, but on microscopic scale it is just a random walk.

Edit: but then perhaps I am thinking about a real process, and you are thinking about a mathematical model.
 
  • #6
lavinia
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I believe in the scale large enough Brownian motions are indistinguishable from (continuous) diffusion, but on microscopic scale it is just a random walk.

Edit: but then perhaps I am thinking about a real process, and you are thinking about a mathematical model.
In general, Brownian motion in mathematics is not necessarily continuous. Sample paths are only continuous almost surely. There is a version of it where the paths are continuous.

As far as real processes are concerned, you do not know whether they are continuous or not since you never have anything except discrete samples of them. For instance, even though stock prices are recorded in discrete jumps, the underlying process may be continuous or may be a continuous time process.
 
  • #7
D H
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Sorry, I didn't understand.
The book I am reading shows Brownian as Markov.
You misunderstood what mathman wrote. Brownian motion is a simple example of a Markov process. He picked one example of a Markov process that is not a Wiener process.

That all Ys are Xs does not necessarily mean that all Xs are Ys.
 
  • #8
Not sure what you mean by a Brownian process but if you mean a Weiner process then there are many Markov processes that are not Weiner processes. For instance,in finance, geometric Brownian motions are commonly use to model securities prices.
sorry I mean Weiner processes (I am assuming that the mathematical treatment of Brownian is Weiner). The text shows that Brownian (with Gaussian distribution & 0 mean) is martingale, and a local martingale is a Brownian (with Gaussian distribution & 0 mean).
Similarly, the text shows that the Brownian (with Gaussian distribution & 0 mean) is a Markov process, now my question is that is it necessary for a Markov process to be a Brownian (with Gaussian distribution & 0 mean). I believe that in your reply you said that it is not necessary for a Markov process to be a Brownian (with Gaussian distribution & 0 mean) & then you have given the example of geometric Brownian motions, am I correct?
 
  • #9
In general, Brownian motion in mathematics is not necessarily continuous. Sample paths are only continuous almost surely. There is a version of it where the paths are continuous.

As far as real processes are concerned, you do not know whether they are continuous or not since you never have anything except discrete samples of them. For instance, even though stock prices are recorded in discrete jumps, the underlying process may be continuous or may be a continuous time process.
what are real processes?
 
  • #10
chiro
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what are real processes?
I think Borek is talking a process that actually exists in nature as opposed to one that is a mathematical curiosity or representation on paper (that either doesn't exist or at hasn't yet been observed).
 
  • #11
I think Borek is talking a process that actually exists in nature as opposed to one that is a mathematical curiosity or representation on paper (that either doesn't exist or at hasn't yet been observed).
thanks
 
  • #12
thanks everyone for very good explanation
 
  • #13
lavinia
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I believe that in your reply you said that it is not necessary for a Markov process to be a Brownian (with Gaussian distribution & 0 mean) & then you have given the example of geometric Brownian motions, am I correct?
Correct. More generally there are Markov processes that are called Ito diffusions that are not Brownian motions. Infinitesimally they look like a Brownian motion multiplied by a function plus a drift term.
 
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  • #14
Correct. More generally there are Markov processes that are called Ito diffusions that are not Brownian motions. Infinitesimally they look like a Brownian motion multiplied by a function plus a drift term.
Sir, thank you so much for great help.

Can you please tell me that what is Geometric Brownian motion, I have tried to read it on google & wiki but unfortunately I can't understand the concept.
 

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