hi guys,i'm trying to solve an assignment problem and just wondering if u guys could tell me if i'm on the right track.given the standard brownian motions, B(s), B(t), B(u), where s < t < u, find the joint pdf.from here find the conditional distribution of P(B(t)|B(s)=a,B(u)=b) and also its mean and variance.now the way i've approached it is:B(s), B(t)-B(s), B(u)-B(t) are all independent with the means 0 and variances s, t-s, u-t respectively.let B(s)=v,B(t)=x,B(u)=yhence,P(B(s), B(t), B(u)) = exp{-(v^2/s + (x-v)^2/t-s + (y-x)^2/u-t)) / {(2pi)^3/2 * sprt{s(t-s)(u-t)}}so to find the conditional probability, i will need to find the joint distribution of B(s) and B(u),P(B(s), B(u)) = exp{-(v^2/s + (y-v)^2/u-s)) / {(2pi) * sprt{s(u-s)}}then substitute B(s)=a, B(u)=b, i.e.P(B(t)|B(s)=a,B(u)=b) = P(B(s)=a, B(t), B(u)=b) / P(B(s)=a, B(u)=b)so what i've done after this is just explode everything and try refactor everything into a normal distribution form. however i can seem to factorize the expression. am i doing the right thing or am i way off?thanks,kefa

how about find the joint density and then integrate

i've already got the joint density.

you're making it way too hard!translate everything to the case where s=0 and a=0.we know the brownian motions at time t and u are jointly normal with mean 0. This means they have covariance matrix C ( t t t u )we can synthesize the Brownian motion at these times by finding a matrix A st C = AA^t, taking a verctor of indep. N(0,1) normals Z =(Z_1,Z_2) , and considering AZ.take the unique upper triangular such A. you can then solve for Z_2 in terms of B_u.we have B_1 in terms of Z_1 and Z_2 and therefore in terms of Z_1 and B_u. We can now read off the distribution of Z_1. I posted on brownian bridging on wilmott a long time ago, Do a search to get.