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Suppose I have a process that follows an arithmetic brownian motion

$dX_t = \sigma dW_t$

How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will "leave" a certain band $\delta$ from the starting time.

I.e, suppose I have a starting point $X_0$ and $t_0$ . Suppose that by $t_1,X_1>X_0+\delta$ or $X_1<X_0−\delta $ . This should add one to the count. Then I want to reset the band to $X_1±\delta$ etc. What are the expected number of times that the process $X_t$ will leave these bands within $\Delta t$?

Otherwise said, if I'm repetitively buying double barrier knock-in options on successive fills on an ABM, how many times should I get hit within a certain time interval.

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  • $\begingroup$ Isn't the probability of getting a crossing related to the crossings within a time period? Because the probability of getting a crossing must be computed with reference to a time period $t$, where $t/\delta t = N$ represents the number of independent increments by the Wiener process. $\endgroup$ Commented Aug 1, 2024 at 8:39

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Heuristically, the time $t$ taken to travel an absolute distance $\delta$ is given by $$\sigma \sqrt{t} = \delta$$. So this gives $t= (\delta/\sigma)^2$ so the number of times this happens in a time period $T$ should be $$\sigma^2 T/\delta^2$$

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  • $\begingroup$ I've seen some really complicated answers that include the pdf of first passage time etc. What is the connection between this answer and those concepts and how are we able to simplify it to this? $\endgroup$ Commented Aug 5, 2024 at 12:17
  • $\begingroup$ I.e. the formula for the pdf of the first passage time from Karatzas and Shreve Second Edition Proposition 6.19 is very complicated, and doesn't seem to have a well defined expectation. How do we know that the above is average time to travel that distance? $\endgroup$ Commented Aug 5, 2024 at 12:38
  • $\begingroup$ Also should there be a $\sqrt{\frac{2}{\pi}}$ here because we are dealing with barriers on both sides? $\endgroup$ Commented Aug 6, 2024 at 9:49
  • $\begingroup$ Those are all fair questions - which is why I wrote ‘heuristically’. $\endgroup$ Commented Aug 6, 2024 at 9:57

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