Exercise $1.$

Variables $x_1$ and $x_2$ follow generalized Wiener processes with drift rates ${}_1$ and ${}_2$ and variances ${}_1^2$ and ${}_2^2.$ What process does $x_1+x_2$ follow if:

$($a$)$ The changes in $x_1$ and $x_2$ in any short interval of time are uncorrelated?

$($b$)$ There is a correlation $$ between the changes in and in any short interval of time?

Exercise $2.$

Consider a variable, $S,$ that follows the process

$dS=dt+dz$

For the first three years, $=2$ and $=3$; for the next three years, $=3$ and $=4.$ If the initial value of the variable is $5,$ what is the probability distribution of the value of the variable at the end of year six?

Exercise $3.$

Stock A and stock B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain your answer.