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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..

Nedellyzer
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1 Answers1

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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 $(B_s-sB_1,B_t-tB_1,B_1)$.

Davide Giraudo
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