Let (B ) and (W ) be two independent Brownian motions and (Y t a B t b W t ) Prove that ( sigma left(Y s , s leq t ight) subset sigma left(B s , W s , s leq t ight) ) and that the inclusion is strict Let (N 1 ) and (N 2 ) be two independent Poisson processes and (Y t a N 1, t ) (b N 2, t ), where (...

To prove the first statement lets denote the sigma algebras as follows sigmaYs s leq t is the sigma ...

Let (B) and (W) be two independent Brownian motions and (Y_{t}=a B_{t}+b W_{t}). Prove that (sigmaleft(Y_{s}, s

Question:

Let $B$ and $W$ be two independent Brownian motions and $Y_{t}=a B_{t}+b W_{t}$. Prove that $\sigma\left(Y_{s}, s \leq t\right) \subset \sigma\left(B_{s}, W_{s}, s \leq t\right)$ and that the inclusion is strict.

Let $N_{1}$ and $N_{2}$ be two independent Poisson processes and $Y_{t}=a N_{1, t}+$ $b N_{2, t}$, where $a eq b$. Prove that $\sigma\left(Y_{s}, s \leq t\right)=\sigma\left(N_{1, s}, N_{2, s}, s \leq t\right)$.

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