... the vector (Wt1 , , Wtn ) has an n-dimensional Gaussian distribution with mean vector and covariance matrix (ti ∧ tj )i,j=1 n (see Exercise 1) So we have to prove that there exist a stochastic ... filtration (Ft )t≥0 and an adapted stochastic process W = (Wt )t≥0 Then W is called a (standard) Brownian motion, (or Wiener process) with respect to the filtration (Ft ) if 16 Stochastic processes ... same mean and covariance function Show that the processes Y and Z defined by Yt = (1 − t)Wt/(1−t) , t ∈ [0, 1), Y1 = and Z0 = 0, Zt = tW(1/t)−1 , t ∈ (0, 1] are standard Brownian bridges 18 Stochastic...