Suppose {B(t)} and {B˜(t)} are two independent standard Brownian motions and ρ is a constant, −1 < ρ < 1. The process Y(t) = ρB(t) + sqrt(1- ρ^2)*B˜(t) is distributed as a normal random variable at every time, Y0 = 0, and it has continuous trajectories. Is {Y(t)} a Brownian motion? (motivate your answer)
There is 5 thing we need to check:
Y0=0 ( which is given in the question)
Independent increments: Y(t1), Y(t2)-Y(t1), ...,Y(tn)-Y(t(n-1)) is independent r.v for any choice of timepoint 0
Stationary increments: Y(t+h)-Y(t) does not depend on t for any h>0
Y(t+h)-Y(t) is N(0,sigma^2*h) for all h>0
Y(t) has cont. trajectories.
I kind of want to say independent and stat increments hold but I cant realy show it.