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alexrem
Topic Author
Posts: 12
Joined: April 3rd, 2009, 4:57 pm

Brownian Motion (Sojourn Time Problem)

Hi, I have been trying this problem for some time now:Given $X(t)=\mu t+\sigma B(t)$ where $B(t)$ is standard Brownian motion, determine the expected amount of time spent in $(0, \delta)$.It is suggested to consider $Y=\int_{0}^{\infty} I{0<X(t)<\delta} dt$ and to find $E(Y)$.I am really not sure where to start, for example how to simplify $\int_{0}^{\infty} I{0<X(t)<\delta} dt$?Thank you.

Alan
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Joined: December 19th, 2001, 4:01 am
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Brownian Motion (Sojourn Time Problem)

Hints:1. Since this is a quant finance forum, think of I(a < X(T) < b) as an option payoff in a Bachelier world (arithmetic BM). What is the (undiscounted) value of the option under your given process? 2. For a general process X(t), express the probability transition density p(t,x,y) as an expectation.
Last edited by Alan on February 28th, 2010, 11:00 pm, edited 1 time in total.

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

Brownian Motion (Sojourn Time Problem)

Last edited by Orbit on March 22nd, 2010, 11:00 pm, edited 1 time in total.

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

Brownian Motion (Sojourn Time Problem)

Last edited by Orbit on March 22nd, 2010, 11:00 pm, edited 1 time in total.

Polter
Posts: 2526
Joined: April 29th, 2008, 4:55 pm

Brownian Motion (Sojourn Time Problem)

Just thinking out loud here -- couldn't one adapt the results for the occupation time (expectation of which seems to be the answer OP is looking for) with doubled local time serving as a density given in Karatzas & Shreve for Brownian Motion (in 3.6)? If so, Tanaka formula might be of help in calculating the expectation (the integral terms may zero out). Some work has to be done to adjust for the drift, but in a later section (3.7) results are given for continuous semimartingales (and X as given by OP is one), so it seems doable. Or is it not the way to go, I'm curious?

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

Brownian Motion (Sojourn Time Problem)

Last edited by Orbit on March 22nd, 2010, 11:00 pm, edited 1 time in total.