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Michael11
science forum Guru Wannabe

Joined: 15 Aug 2005
Posts: 103

Posted: Wed Jul 19, 2006 7:16 am    Post subject: Is there a way to write out the process of the cumulative minimum of a Brown Motion process?

Hi all,

I have a Geometric Brown Motion process Xt,

dXt=mu*Xt*dt + sig*Xt*dWt,

where Wt is a Brownian Motion.

Define Mt=cumulative minimum of Xt = min(Xs, s from 0 to t).

Is there a way to find out the formulation of the process of Mt?

Hopefully it will have some nice formula similar to dXt above?

Thanks a lot!
matt271829-news@yahoo.co.
science forum Guru

Joined: 11 Sep 2005
Posts: 846

Posted: Thu Jul 20, 2006 1:36 pm    Post subject: Re: Is there a way to write out the process of the cumulative minimum of a Brown Motion process?

Michael wrote:
 Quote: Hi all, I have a Geometric Brown Motion process Xt, dXt=mu*Xt*dt + sig*Xt*dWt, where Wt is a Brownian Motion. Define Mt=cumulative minimum of Xt = min(Xs, s from 0 to t). Is there a way to find out the formulation of the process of Mt? Hopefully it will have some nice formula similar to dXt above? Thanks a lot!

Sorry if this appears more than once. Google Groups is being very flaky
right now.

This may or may not be of any use. There's a formula at
http://www.risklatte.com/brownianMotion/brownian003.php which claims to
give the probability of a variable following geometric brownian motion
hitting a barrier sometime between time t = 0 and t = T.

In the formula, I think

Their S is your X(0) (i.e. the value of the random variable at time
zero).
Their X is the barrier, where X < S.
N(z) is the integral from -infinity to z of a standard normal
distribution.
I'm GUESSING that their mu is equal to your mu - sigma^2/2 (this
adjustment is because the drift of log(X) is not actually equal to mu,
IIRC).

This formula seems to check out against random simulations, but I
haven't attempted to work out the derivation, and the reference doesn't
give one. Probably if you scour the vast literature for derivatives
pricing you will find one somewhere.

Then, the probability of hitting the barrier is the same as the
probability that the minimum between t = 0 and t = T is less than or
equal to the level of the barrier. Thus the formula gives you the
cumulative distribution of the minimum, from which you can derive the
PDF of the minimum. I'm not sure how you would derive a process for the
minimum from this though.

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