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Confusing Property of Brownian Motion

E

EdwardT

Member
Hi

On pp263 of 7th ed Hull (Section 12.2 - Continuous-time Stochastic Processes) it has:

"2. The expected number of times z equals any particular value in any time interval is infinite."

(where z is standard brownian motion)

My understanding of this is:

"Given any value, brownian motion will hit this value infinitely often, no matter what the time scale."

Is my understanding correct?

If so, doesn't this property completely invalidate the use of Geometric Brownian motion as a model for the stock market as it would basically say that every day the stock market should take every value in (0,infinity)?

The only reason I'm asking is I'm sure I've seen an Acted CT8 question in the past that said something similar.

Equally, I'm sure I've seen maths papers that derive the distribution of first hitting times for Brownian motion.

Finally, I could understand if the statement actually meant to say:

"Given that the Brownian motion currently takes a value z_0, the expected number of times it takes the value z_0 in any time interval is infinite."
 
I think it refers to the current value. ie Brownian motion will cross the current value infinitely many times in a given time period
 
Thanks, that's what I was thinking.

Normally I'd ignore it but I was sure I'd seen an Acted example in CT8 (that I ignored & now can't find) that used the time-inversion property of Brownian motion to demonstrate the more surprising interpretation of Hull.
 
It seems to be implied that the process has already hit the particular value of z at least once.

I.e. the hitting time to a particular value of z depends on the starting value. If the starting value is z, the hitting time is zero and the expected number of times the process equals z is infinite in the next time interval however short this is.

If the starting value is not z, then it is not certain that z will be hit in a given time interval, although if it is then the value z will be equalled infinitely many times. The likelihood of this depends on the nature of the process, including the drift, and the starting value.
 
Yes, I think Hull's wording is a bit "loose" here. I think what he means is "The expected number of times z equals any particular value THAT IT HAS HIT in any time interval is infinite". The way this is usually expressed is to say that once it hits a particular value it will hit it again infinitely often in the next instant, however short that "instant" is.
 
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