I'm wondering if anyone would be able to help me on the following two problems:
Let $X_n$, n=0,1,2,..., be a random walk on (0,1,...,N). Assume that starting from any i it is a martingale. Prove that p(0,0) = p(N,N) = 1.
Lognormal stock prices: Consider the special case of Example 5.4 (below) in which $X_i$ = $e^η$ where $η_i$ = normal(μ, $σ^2$). For what values of μ and σ is $M_n$ = $M_0$ * $X_1$...$X_n$ a martingale?
*Example 5.4: (Products of Independent Random Variables). To build a discrete time model of the stock market we let $X_1$,$X_2$,... be independent ≥ 0 with E$X_i$=1. Then $M_n$=$M_0$ * $X_1$...$X_n$ is a martingale with respect to $X_n$. To prove this we note that E($M_n$$_+$$_1$) - $M_n$ | $A_v$)= $M_n$E($X_n$$_+$$_1$ - 1 | $A_v$) = 0
For #1, I know we can set E$X_n$=E$X_0$, and I was thinking that if we start at i then E$X_0$ is i. I'm not sure how to put these thoughts together in a proof.
For #2, putting it all together, it's 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 have no idea where to start.