Let $\{X(t), t \geq 0\}$ be a $BM(5,7)$ (Brownian Motion) with $X(0) = 4$. Let $Y = X(5) - X(3)$ and $Z = X(15) - X(10)$. Find the join probability $f_{Y,Z}(y,z)$.
I know that $P(Y,Z| X(0) = 4) = P(Y|X(0) = 4)\cdot P(Z|X(0)=4)$ since the increments are independent. However, I'm not sure how to find the distribution of an increment condition on $X(0) = 4$.
