Q$.3($Brownian gymnastics, $20pts).$ Let $(W_t{)}_{t}0$ be a standard Brownian motion. Let

$B_0$inR,$inR,$ and $>0.$ Consider the process $(B_t{)}_{t}0$ defined by

$B_t=B_0+t+W_t,t0.$

$(B_t{)}_{t}0$ is called an arithmetic Brownian motion with initial value $B_0,$ drift $,$ and

volatility $.$

a$.(5)$ Let $t0.$ What is the distribution of $B_t?$

b$.(5)$ Let $t,s0.$ Compute Cov$(B_t,B_s).$

c$.(5)$ Compute the moment generating function of $W_1,$ that is$,$ compute $E[e^{r{W}_1}]$ for

every rinR.

d$.(5)$ Let $t0.$ Compute the moment generating function of $B_t,$ that is$,$ compute

$E[e^{r{B}_t}]$ for every rinR.