Exercise 4 (26 points) Recall that since the paths of Brownian motion are continuous it is possible to consider Xt=0tBudu as a Riemann integral, that is, Xt=limni=0n1ntBti. (i) (6 pts) Compute E[Xt]. (Hint: dominated convergence theorem and maxu[0,t]Bu has finite moments by Bachelier theorem.) (ii) (6 pts) Compute Var[Xt]. (iii) (6 pts) What is the distribution of i=0n1ntBti for each n1 ? Would you expect the distribution of Xt to be of the same type? Discuss. (iv) (8 pts) Apply Ito's formula to f(t,Bt)=tBt and deduce the distribution of Xt