# ABM Crossing Times

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 $$X_1 . This should add one to the count. Then I want to reset the band to $$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.

• 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. Commented Aug 1 at 8:39

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 $$\sigma^2 T/\delta^2$$
• Also should there be a $\sqrt{\frac{2}{\pi}}$ here because we are dealing with barriers on both sides? Commented Aug 6 at 9:49