Verifying the martingale property from transition probabilities of a Markov chain

Question

Let $X_n$ be a Markov chain whose transition probabilities are $P_{i,j}=1/[e(j-i)!]$ for $i=0,1,...$ and $j=i,i+1,...$.

Verify the martingale property for:

(a) $Y_n=X_n-n$;

(b) $U_n=Y_n^2-n$;

(c) $V_n=exp\{X_n-n(e-1)\}$.

For a process to be a martingale, we have to show that $E[X_{n+1}|\mathcal{F}_n]=X_n$. However, most of the problems I've verified to be a martingale for have been by using algebraic expansion and simplification. I am not sure how to do it using transition probabilities. Can I get some examples or pointers of how to do this? Thanks

Answer 1

For (a), you should have the term $E[X_{n+1} \mid F_n]$. This can be expanded as $\sum_{j=X_n}^\infty j \cdot P_{X_n, j}$. Then apply the definition of $P_{i,j}$ and you'll get an infinite series. Factor out $1/e$ and change the bounds to start at zero and you should notice it looks similar to the exponential infinite series definition. Sincerely, another 491 student.