This expectation troubles me because it includes both terminal and minimum value within the indicator function.Anyone has some hints?B(t) is a standard Brownian Motion. How to calculate the following expectation:a is constant and constants K2<K1<0.Thanks very much for that!

You can turn it into a conditional expectation. Then transform to the integral with PDF inside.In order to get the PDF, you calculate its CDF first. Using reflection principle to get the joint probability.

Got it!Thanks very much for that!

QuoteOriginally posted by: weekendchenThis expectation troubles me because it includes both terminal and minimum value within the indicator function.Anyone has some hints?B(t) is a standard Brownian Motion. How to calculate the following expectation:a is constant and constants K2<K1<0.Thanks very much for that!your expectation does not contain K_1 and K_2 though one could suppose that they are in min expression. There is a book of 2 French women. One is Dana translated by Kennedy about Finance mathematics. They presented formulas for joint densities of max , min and final value of the Weiner process. The book is

QuoteOriginally posted by: listQuoteOriginally posted by: weekendchenThis expectation troubles me because it includes both terminal and minimum value within the indicator function.Anyone has some hints?B(t) is a standard Brownian Motion. How to calculate the following expectation:a is constant and constants K2<K1<0.Thanks very much for that!your expectation does not contain K_1 and K_2 though one could suppose that they are in min expression. There is a book of 2 French women. One is Dana translated by Kennedy about Finance mathematics. They presented formulas for joint densities of max , min and final value of the Weiner process. The book is List. Thanks for your reply!It has a mistake before. Now it is corrected, containing both K1 and K2.

I missed something when I read the formula. All constants where written initially correctly.

Somebody please put weekendchen out of his misery (I think he has asked the same question in 3 threads)and latex the joint density of (B_T, m_T), the BM and its minimum.

Hey Alan. Thanks for your words.Yes, I have asked and posted 3 related topics. They are similar but not totally the same. There are 2 topics that look the same (with the same equation). That is because I copy the Latex Equation from one to another one and would like to do some change. But then when one is changed, another one change automatically. I have tried to make it different, but not succeed.But the good news is that, now I have figured out all 3 topics, with the help of you and other online friends, especially foxkingdom. I learn a lot from him

Well, good. As this is a FAQ, maybe you can latex the joinf pdf of (B_T, m_T) for future readers of this thread.

QuoteOriginally posted by: AlanWell, good. As this is a FAQ, maybe you can latex the joinf pdf of (B_T, m_T) for future readers of this thread.Latex the mirror principal one, mine is too twisted...

According to Karatzas and Shreve, page 95:the joint density of B_t (the Brownian motion level at time t) and the running maximum M_t over [0,t] is for t> 0For the running minimum (rather than maximum) I suppose we would take minus the maximum of -B_t giving:

QuoteOriginally posted by: acastaldoAccording to Karatzas and Shreve, page 95:the joint density of B_t (the Brownian motion level at time t) and the running maximum M_t over [0,t] is for t> 0For the running minimum (rather than maximum) I suppose we would take minus the maximum of -B_t giving:it looks correct bearing in mind that B_t has the same distribution as - B_t and min for B will be convertedto max for - B

Thanks, guys. Of course, the support of the densities is {B <= M && M >=0} and {B >= m && m <= 0}, respectively.