Background:

For 0s1<s2, we recall that the pair(Ws1,Ws2) is a centered Gaussian vector with variance matrix (s1s1s1s2), and we verify therefore thatWs1Ws2 is also Gaussian with characteristicsE[Ws1Ws2]=s2s1Ws2 and Var[Ws1Ws2]=s1(1s2s1). Denote s=2s1+s2, we have thatWs(Ws1=x1,Ws2=x2) has a Gaussian distribution with conditional meanx=2x1+x2 and conditional variance 4s2s1. We can justify that conditional distribution ofWs(Ws1=x1,Ws2=x2,(Wu)u/[s1,s2]) is N(x,4s2s1).

Use Python to answer the following question:

For a positive integer n, denote T=2nT,tin=iT,i=0,...,2n. Our objective is to simulate a discretization of a Brownian motion W. Use the last property to simulate backward the discretized Brownian motion: start by drawing copies ofWT, then WT/2=Wt11, thenWT/4=Wt12 and W3T/4=Wt32, etc... All implementations should be run with the value T=10. Then, compute the corresponding sample mean and variance of WT, and the sample covariance of (WT,WT/2). Comment on the results by varying the value of n.