PROBLEM
$U_t = B_t – tB_1$, $B_t$ is a Brownian motion on $[0,1]$.
What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$.
I think that a Brownian Bridge is a Brownian motion between $[0,1]$, but I'm not sure, if I am wrong please tell me.
For the second part I know that $E(U_t) = 0$ and $Var(U_t) = t$
Also, $E(U_s) = 0$ and $Var(U_s) = s$.
However, I don't know how to give the total distribution with the covariance of this vector.
I hope somebody can help me..
Best Answer
Brownian bridge is a process whose (almost surely) continuous paths join $(0,0)$ and $(1,0)$ (that's the explanation for the term bridge).
For the distribution of $(U_s,U_t)$, notice that we know the joint distribution of $(B_s,B_t,B_1)$. Hence using a substitution, we can deduce those of $(B_s-sB_1,B_t-tB_1,B_1)$.