I am questioning if anybody would be capable of assist me on the next two issues:
Let $X_n$, n=0,1,2,…, be a random stroll on (0,1,…,N). Assume that ranging from any i it’s a martingale. Show that p(0,0) = p(N,N) = 1.
Lognormal inventory costs: Take into account the particular case of Instance 5.4 (under) by which $X_i$ = $e^η$ the place $η_i$ = regular(μ, $σ^2$). For what values of μ and σ is $M_n$ = $M_0$ * $X_1$…$X_n$ a martingale?
*Instance 5.4: (Merchandise of Impartial Random Variables). To construct a discrete time mannequin of the inventory market we let $X_1$,$X_2$,… be impartial ≥ Zero with E$X_i$=1. Then $M_n$=$M_0$ * $X_1$…$X_n$ is a martingale with respect to $X_n$. To show this we word that E($M_n$$_+$$_1$) – $M_n$ | $A_v$)= $M_n$E($X_n$$_+$$_1$ – 1 | $A_v$) = 0
For #1, I do know we will set E$X_n$=E$X_0$, and I used to be pondering that if we begin at i then E$X_0$ is i. I am undecided easy methods to put these ideas collectively in a proof.
For #2, placing all of it collectively, it is asking for what values of μ and σ is $M_n$ = $M_0$∏$e^η$$^i$ (from i=1 to n) a martingale? For this one I do not know the place to begin.