2. Let (0, F, P) be a probability space. Let {n)>, be a filtration. a. Let Mn denote the symmetric random walk with Zn = Mn - Mn-1 i. Calculate g(1) = E[edZ~]. ii. Let Yn = exp{AMn - nlogg(X)}. Show that E[Y,+1|Fa] = Yn. Trick for expential martingale: is to consider E ntly, Fr iii. Let Yn = Mn -n Show that E[Y,+1|Fa] = Yn. (Yn is a martingale.) iv. Let Yn = Mi - 3nMn Show that E[Y,+1|Fa] = Yn. (Yn is a martingale). v. Suppose 0 E R and let Yn = (secho)neoMn n = 0,1,2, .... Show that E[Y,+IF] = Yn. (Y, is a martingale). (Hint: Use the trick and sinh(z) = =)