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."
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."