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guillaume07
Topic Author
Posts: 37
Joined: February 11th, 2007, 8:52 am

### Brownian Motion

Hello,I don't understant a propriety of wiener process. I don't understand, When one said , that wiener process has independent increment.let W be a wiener process and t1 and t2 two instant of this process ,I know that W(t2)-Wt(1) is independant of W(t1)-W(t0). but indeed i don't really understand why this property is precisely let like that. Why is it minus and not add or a thing like that.In an interview someone ask me , W(t1) is independant of what ? I did not succeed to answer it

Zedr0n
Posts: 244
Joined: April 6th, 2007, 5:07 am

### Brownian Motion

Let Wt be the process of the stock price, then for example the up-move of the stock price doesn't depend on the value of the stock at time t. It also allows for another handy property - markov property of brownian motion. For example, the process W(t+a) - W(a) for a constant a is brownian motion - so, generally it's related to market efficiency - all the info is in the price so future price behavior isn't dependent on the past prices.As for W(t1), it is independent of every (W(t3) - W(t2)) where (0,t) and (t2,t3) don't overlap.I don't really know - the point in mathematics is to build the simplest possible model with good approximation, the brownian motion seems to fit, and independent increments are a big plus
Last edited by Zedr0n on April 7th, 2007, 10:00 pm, edited 1 time in total.

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

### Brownian Motion

QuoteOriginally posted by: Zedr0nindependent increments are a big plusone such advantage is that you don't have to know the history from the beginning of times. in brownian motion processes, all you you need to know is where is it now in order to get an idea of what's going to happen in future. in more general case of markovian processes, you may need to know a little bit of history, maybe a couple of steps back, but the point is that the memory is short.

guillaume07
Topic Author
Posts: 37
Joined: February 11th, 2007, 8:52 am

### Brownian Motion

thanks

letiand
Posts: 11
Joined: April 3rd, 2007, 1:58 am

### Brownian Motion

QuoteOriginally posted by: jawabeanQuoteOriginally posted by: Zedr0nindependent increments are a big plusone such advantage is that you don't have to know the history from the beginning of times. in brownian motion processes, all you you need to know is where is it now in order to get an idea of what's going to happen in future. in more general case of markovian processes, you may need to know a little bit of history, maybe a couple of steps back, but the point is that the memory is short.As I know, the definition of markovian processes says that the future state depends only upon the present state and is conditionally independent of the past states.Why we need to know a liittle bit of history? THanks...

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

### Brownian Motion

QuoteOriginally posted by: letiandAs I know, the definition of markovian processes says that the future state depends only upon the present state and is conditionally independent of the past states.Why we need to know a liittle bit of history? THanks...i meant higher order markovian processes like moving average processes. if you extend the present to a few steps back, then you can still call such processes markovian.

letiand
Posts: 11
Joined: April 3rd, 2007, 1:58 am

### Brownian Motion

QuoteOriginally posted by: jawabeanQuoteOriginally posted by: letiandAs I know, the definition of markovian processes says that the future state depends only upon the present state and is conditionally independent of the past states.Why we need to know a liittle bit of history? THanks...i meant higher order markovian processes like moving average processes. if you extend the present to a few steps back, then you can still call such processes markovian.Thanks...

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