Digitals and Brownian bridge again...

Clopinette · July 28th, 2005, 1:50 pm

Could someone correct me here:Say you have a brownian motion W(t). You have simulated its values in a montecarlo for dates T1 and T2.What is the law of "W(u) conditionally to W(T1) and W(T2)" for a date u in the interval [T1; T2]?I am no too sure, is it a gaussian law with : - mean = p.W(T2) + (1-p).W(T1) - Variance p.(1-p).(T2 - T1)/T1Here I took p = (u-T1)/(T2-T1).I am told that the variance scale is wrong.

JadePoisson · July 28th, 2005, 1:58 pm

It is a standard BM bridge, just consider it as BM starting at T1 and B(T1), everything else is standard.

Clopinette · July 28th, 2005, 2:06 pm

Thanks JadePoissonBut precisely all the docs I have about bb today with me take the particular case of T1 = 0. That is why I am not too sure about the scale of my variance.Do you think my variance looks right? Or should I take Variance = p.(1-p).(T2 - T1)And what do you think of p?

JadePoisson · July 28th, 2005, 2:52 pm

Just because BM is (strongly) Markovian, B(s)-B(T1) follows a standard BM starting with zero... and thus B(T_2)-B(T1) replaces the original terminal pt... I leave the computaion to you and I need to run for my apartment.......(huge headache for me)

Clopinette · July 28th, 2005, 2:58 pm

I think I got you JadePoisson, you are a star.Thanks and good luck with your flat

Borya · July 28th, 2005, 8:01 pm

This formula is in Peter Jaeckel's book "Monte-Carlo Methods in Finance", p.125,derivation on page 138.Formula for variance is p*(1-p)*(T2-T1)