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EdwinAfitile
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Is Brownian Motion Ergodic?

September 17th, 2016, 12:26 pm

 Friends, those who are into these things. I am thinking that Brownian Motion is Markov & is also a Martingale but I am struggling to conclude about the Ergodicity. If Brownian motion is not Ergodic, then we need to rethink a suitable model for stock prices when considering longer time frames. This can have an effect on the pricing of Derivatives, for example the prices of Out of the Money Put Options can be zero.

Any takers?
 
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list1
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Re: Is Brownian Motion Ergodic?

September 17th, 2016, 12:48 pm

one characteristic of the ergodic property of a random function is an equality means over the time and space. If observations do not present closeness means it looks difficult to expect ergodicity.
 
EdwinAfitile
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Re: Is Brownian Motion Ergodic?

September 17th, 2016, 7:00 pm

one characteristic of the ergodic property of a random function is an equality means over the time and space. If observations do not present closeness means it looks difficult to expect ergodicity.
Maybe it will be interesting to consider if all Markov processes are ergodic?
 
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list1
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Re: Is Brownian Motion Ergodic?

September 18th, 2016, 1:19 pm

one characteristic of the ergodic property of a random function is an equality means over the time and space. If observations do not present closeness means it looks difficult to expect ergodicity.
Maybe it will be interesting to consider if all Markov processes are ergodic?
 "if all Markov processes are ergodic" sounds a little bit strange. Markov processes is a big class of random processes. If some conditions are satisfied a Markov process is ergodic. It was a book about stability stochastic processes written in about 1970 by Khasminskii. It wa a chapter where Ergodic properties of SDE solutions were studied.
 
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bearish
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Re: Is Brownian Motion Ergodic?

September 18th, 2016, 1:46 pm

Welcome, Edwin! I am glad you and list found each other so quickly. Could you expand slightly on your opening remark: "This can have an effect on the pricing of Derivatives, for example the prices of Out of the Money Put Options can be zero"?
 
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list1
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Joined: July 22nd, 2015, 2:12 pm

Re: Is Brownian Motion Ergodic?

September 18th, 2016, 2:20 pm

Welcome, Edwin! I am glad you and list found each other so quickly. Could you expand slightly on your opening remark: "This can have an effect on the pricing of Derivatives, for example the prices of Out of the Money Put Options can be zero"?
bearish, the question was about "Brownian Motion is Markov & is also a Martingale but I am struggling to conclude about the Ergodicity " is about other drama.
By the way if you are interested when " Put Option can be zero" I would say that "Out of the Money" does not relevant observation for you statement. Put Options are equal to zero for the market scenarios which implies zero put payoff. In BS pricing they deal with expected values and ignore the fact that expression under expectation also has a financial nature.
 
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Gamal
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Re: Is Brownian Motion Ergodic?

September 18th, 2016, 6:41 pm

Follow this fascinating thread. Or threat, I'm not sure.
 
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outrun
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Re: Is Brownian Motion Ergodic?

September 18th, 2016, 7:07 pm

Welcome, Edwin! I am glad you and list found each other so quickly.
I think we must applaud list1 for that.

I do finds things a bit confusing though every now and then.
 
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acastaldo
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Re: Is Brownian Motion Ergodic?

September 19th, 2016, 3:18 pm

Is Brownian Motion Ergodic?
If only someone could invent a worldwide, searcheable network which stored published papers by academic luminaries and was readily accessible to every student or professional in the world. Then it would be easy to answer such questions.

In the absence of this, perhaps you could write a letter to the family of the late Gopinath Kallianpur (1925-2015). I heard he worked on this question in his youth and perhaps he passed on his conclusion to friends and relatives who may be able to help you.
 
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katastrofa
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Re: Is Brownian Motion Ergodic?

September 19th, 2016, 3:43 pm

The Brownian motion is ergodic...
 
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list1
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Joined: July 22nd, 2015, 2:12 pm

Re: Is Brownian Motion Ergodic?

September 19th, 2016, 3:54 pm

I think the best reference related to ergodic properties of the SDE solutions is 
 
EdwinAfitile
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Joined: September 5th, 2016, 12:22 pm

Re: Is Brownian Motion Ergodic?

September 19th, 2016, 4:13 pm

I think the cleaner question is to ask if stock prices observed in the markets exhibit any ergodicity.
 
EdwinAfitile
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Re: Is Brownian Motion Ergodic?

September 19th, 2016, 4:25 pm

I am looking for a model that can be able to capture the long term benhaviour of stock markets
 
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list1
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Joined: July 22nd, 2015, 2:12 pm

Re: Is Brownian Motion Ergodic?

September 19th, 2016, 5:24 pm

I think the cleaner question is to ask if stock prices observed in the markets exhibit any ergodicity.
It is more simpler question can be ask for the GBM stock model. No one care about testing. It is a basic assumption. If you can see any evidence of ergodicity in the market you can assume it if not such assumption better to reject. Long run stock market behavior much worse be predicted than weather in 5 years forward.
 
EdwinAfitile
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Joined: September 5th, 2016, 12:22 pm

Re: Is Brownian Motion Ergodic?

September 19th, 2016, 5:33 pm

 Hi List, thanks for your answers. What other language do you speak?
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