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rudinable
rudinable
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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.

I've seem some other related content online but it typically deals with the probability of getting a crossing etc, but not the crossings within a time period.

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.

I've seem some other related content online but it typically deals with the probability of getting a crossing etc, but not the crossings within a time period.

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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rudinable
rudinable
  • 111
  • 3

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<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.

I've seem some other related content online but it typically deals with the probability of getting a crossing etc, but not the crossings within a time period.